Tatsuya Shirakawa, profile picture
Uploaded byTatsuya Shirakawa
PDF, PPTX2,580 views

Hyperbolic Neural Networks

The document discusses hyperbolic neural networks, emphasizing the integration of Möbius gyrovector space and Riemannian hyperbolic geometry. It covers various neural network components and their construction in hyperbolic space, showcasing experimental results that support the methods' efficiency. Additionally, it contrasts hyperbolic and Euclidean spaces, including insights on metrics, models, and transformations relevant to hyperbolic geometry.

Technology◦
Related topics:
Deep Learning•

Embed presentation

Download as PDF, PPTX
Hyperbolic Neural Networks 2019 Jan. 26th Tatsuya Shirakawa (ABEJA, Inc.)
ABEJA, Inc. (Researcher) - Deep Learning (CV, Graph, NLP, ) - Machine Learning - Mathematical Optimization - https://github.com/TatsuyaShirakawa Tech blog http://tech-blog.abeja.asia/ Poincaré Embeddings Graph Convolution Annotation Self Intro. Hyperbolic
This paper From my blog article
Paper Intro. O. E. Ganea+, “Hyperbolic Neural Networks”, NeurIPS2018 (spotlight) • Exploring the connection between 
 Möbius gyrovector space and Riemannian hyperbolic geometry • Construction of several components of neural networks in hyperbolic space • Experiments which prove the efficiency of the proposed construction
Methods in Hyperbolic Spaces [M. Nickel+]
 Poincaré Embeddings ~ 17’ 05 18’ 04 [C. D. Sa+]
 Representation Tradeoff ~
 (Near-exact tree embs., h-MDS) [O. Ganea+]
 Hyperbolic Entailment Cones
 (Poincare embs. + Order embs.) 18’ 05 [C. Gulcehre+]
 Hyperbolic Attention Networks
 (Einstein Mid. Point) [O. Ganea+]
 Hyperbolic Neural Networks [M. Nickel+]
 Learning Continuous Hierarchies
 in the Lorentz Model~ [A. Tifrea+]
 Poincaré Glove ~ (Poincaré Glove) 18’ 06 18’ 10 …
Embeddings [M. Nickel+]
 Poincaré Embeddings ~ 17’ 05 18’ 04 [C. D. Sa+]
 Representation Tradeoff ~
 (Near-exact tree embs., h-MDS) [O. Ganea+]
 Hyperbolic Entailment Cones
 (Poincare embs. + Order embs.) 18’ 05 [C. Gulcehre+]
 Hyperbolic Attention Networks
 (Einstein Mid. Point) [O. Ganea+]
 Hyperbolic Neural Networks [M. Nickel+]
 Learning Continuous Hierarchies
 in the Lorentz Model~ [A. Tifrea+]
 Poincaré Glove ~ (Poincaré Glove) 18’ 06 18’ 10 …
Neural Network Construction [M. Nickel+]
 Poincaré Embeddings ~ 17’ 05 18’ 04 [C. D. Sa+]
 Representation Tradeoff ~
 (Near-exact tree embs., h-MDS) [O. Ganea+]
 Hyperbolic Entailment Cones
 (Poincare embs. + Order embs.) 18’ 05 [C. Gulcehre+]
 Hyperbolic Attention Networks
 (Einstein Mid. Point) [O. Ganea+]
 Hyperbolic Neural Networks [M. Nickel+]
 Learning Continuous Hierarchies
 in the Lorentz Model~ [A. Tifrea+]
 Poincaré Glove ~ (Poincaré Glove) 18’ 06 18’ 10 …
Hyperbolic Space • Manifolds with constant sectional curvature
 - zero curvature: Euclidean space
 - pos. curvature: hypersphare
 - neg. curvature: hyperbolic space • Equivalent models of hyperbolic space
 - the upper half-space model
 - the poincaré disk model
 - the Beltrami-Klein model
 - the Lorentz model (hyperboloid model)
 …
Poincaré Disk/Ball Model • A model for hyperbolic space, which is suitable for visualization • A unit-Ball equipped with the following Riemannian metric • hyperbolic 
 The volume of a ball grows 
 exponentially with the radius
 tree 
 the number of nodes grows 
 exponentially with the depth 「異空間散歩！双曲空間を歩いてみよう。」
 (http://nunuki.hatenablog.com/entry/2018/12/15/055136)
Euclidean vs Hyperbolic Euclidean Hyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Gyrovector Space
Euclidean vs Hyperbolic Euclidean Hyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b Möbius transform, relection Algebraic structure Vector Space Gyrovector Space
Euclidean vs Hyperbolic Euclidean Hyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Gyrovector Space M¨obius transformation in 2d (with polar decomp.) az + b cz + d = ei✓ w(z), w(z) = z + w ¯wz + 1 = · · · = (1 + 2w · z + kzk2 )w + (1 kw2 k2 )z 1 + 2w · z + kwk2kzk2 <latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit><latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit><latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit><latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit>
Euclidean vs Hyperbolic Euclidean Hyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Gyrovector Space z M¨obius transformation a(x) = (1 + 2a · x + kxk2 )a + (1 kak2 )x 1 + 2a · x + kak2kxk2 or, with scale parameter c > 0, c a(x) = (1 + 2ca · x + ckxk2 )a + (1 ckak2 )x 1 + 2ca · x + c2kak2kxk2 <latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">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</latexit><latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">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</latexit><latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">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</latexit><latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">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</latexit>
Euclidean vs Hyperbolic Euclidean Hyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Möbius Gyrovector Space vector M¨obius gyrovector x 7! x + a x 7! a x := a(x) = a x(0) x 7! rx x 7! r ⌦ x (continuation of n x = x x · · · x)<latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit>
More Gyrovector space • Gyrovector space can be also defined for other hyperbolic models
 - Einstein gyrovector space <—> algebraic formulation of special relativity • Relation to Riemannian hyperbolic geometry
 - • Mid point • Gyrotrigonometry
 … a ( a b) ⌦ t ! geodesic from a to b<latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit><latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit><latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit><latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit>
Riemannian Hyperbolic Geometry
 exp/log map • exp: map from tangent space to manifold • log: reversed map from manifold to tangent space • In hyperbolic space, exp/log map 
 extend to whole space, having 
 closed form of
Hyperbolic Neural Networks Euclidean Hyperbolic Softmax —
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">AAACYnicbVFNa+MwFJTdj816+5E0x92D2LCQUgh2KLSXQuhe9lLowiYpxMHIynMiKstGel5qTP5kbz310h+ycupDtu0DwWjePI00inMpDPr+k+Pu7O7tf2p99r4cHB4dtzsnE5MVmsOYZzLTdzEzIIWCMQqUcJdrYGksYRrf/6z707+gjcjUHyxzmKdsqUQiOENLRe2yfxMFIb2Jhqc0lJBg3wtjWApVMa1Zua6kXHthynAVJ9XD2mrD7e3QC0EtGq0XarFc4Sm9sucFdHuKntUW29Qwavf8gb8p+h4EDeiRpm6j9mO4yHiRgkIumTGzwM9xbr1RcAnWvTCQM37PljCzULEUzLzaRLSmPyyzoEmm7VJIN+z2RMVSY8o0tsr6juZtryY/6s0KTC7nlVB5gaD4q1FSSIoZrfOmC6GBoywtYFwLe1fKV0wzjvZXPBtC8PbJ78FkOAj8QfD7vDe6buJoka/kO+mTgFyQEflFbsmYcPLs7DlHzrHz4npux+2+Sl2nmemS/8r99g+OAbTf</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit>
Hyperbolic Neural Networks Euclidean Hyperbolic Softmax —
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit> “Hyperplane margin classifiers on the multinomial manifold”, NIPS2014 Euclidean ak · x bk = ak · (x pk) (ifbk = ak · pk) = kakksign(ak · (x pk))d(x, {y|ak · (y pk) = 0}) Hyperbolic q gp(ak, ak)sign(ak · (x pk))d(x, {y|ak · (y pk) = 0}) <latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4</latexit><latexit sha1_base64="mXej7eo96vAzELZ/uggt3i9doxs=">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</latexit><latexit sha1_base64="mXej7eo96vAzELZ/uggt3i9doxs=">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</latexit><latexit sha1_base64="emtCgJQUpEekBRkSlrXZJOYv+Cc=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit>
Hyperbolic Neural Networks Euclidean Hyperbolic Softmax —
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">AAACYnicbVFNa+MwFJTdj816+5E0x92D2LCQUgh2KLSXQuhe9lLowiYpxMHIynMiKstGel5qTP5kbz310h+ycupDtu0DwWjePI00inMpDPr+k+Pu7O7tf2p99r4cHB4dtzsnE5MVmsOYZzLTdzEzIIWCMQqUcJdrYGksYRrf/6z707+gjcjUHyxzmKdsqUQiOENLRe2yfxMFIb2Jhqc0lJBg3wtjWApVMa1Zua6kXHthynAVJ9XD2mrD7e3QC0EtGq0XarFc4Sm9sucFdHuKntUW29Qwavf8gb8p+h4EDeiRpm6j9mO4yHiRgkIumTGzwM9xbr1RcAnWvTCQM37PljCzULEUzLzaRLSmPyyzoEmm7VJIN+z2RMVSY8o0tsr6juZtryY/6s0KTC7nlVB5gaD4q1FSSIoZrfOmC6GBoywtYFwLe1fKV0wzjvZXPBtC8PbJ78FkOAj8QfD7vDe6buJoka/kO+mTgFyQEflFbsmYcPLs7DlHzrHz4npux+2+Sl2nmemS/8r99g+OAbTf</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit>
Hyperbolic Neural Networks Euclidean Hyperbolic Softmax — 
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit> extend
Hyperbolic Neural Networks Euclidean Hyperbolic Softmax — 
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit>
Hyperbolic RNN/GRU • Hyperbolic RNN • Hyperbolic GRU

Experiments
Task1 — SNLI • SNLI (Stanford Natural Language Inference) — textual entailment
 
 - sentence A: “Little kids A. And B. Are playing soccer.”
 - sentence B: “Two children are playing outdoors.”
 
 Q) A => B?
Task2 — PREFIX-Z% • PREFIX-Z% (Z=10, 30, 50) — PoC task
 
 - sent1 (length=n1) = random seq. of symbols (words)
 - sent2 (length=n2) = random seq. of symbols (len(sent2) <= len(sent1))
 - sent1’(length=n2) = take first n2 symbols of sent1 and replace Z% symbols with randomized ones
 
 Q) detect whether (s1, s2) comes from (sent1, sent2) or (sent1, sent1’)
 (s1, s2) Label s1 = 23, 42, 52, 29, 20, 22, 22, 11, …, 60, 16
 s2 = 58, 17, 100, 27, 10, 37, 24, 50 0 s1 = 23, 42, 52, 29, 20, 22, 22, 11, …, 60, 16
 s2 = 66, 42, 52, 29, 20, 22, 38, 11 1
Results
Decision Boundary (SNLI) • No advantage for hyperbolic MLR in SNLI task
MLR classification experiments Poincaré Embs.
 (https://github.com/facebookresearch/poincare-embeddings) Take a subtree
 A = node in the subtree
 B = node not in the subtree Classify with - hyperbolic MLR - Euclidean MLR - log_0 + Euclidean MLR
More Details and Related Works https://tech-blog.abeja.asia/entry/hyperbolic_ml_2019

More Related Content

PPTX
Graph Neural Networks
bytm1966
 
PDF
ドメイン適応の原理と応用
byYoshitaka Ushiku
 
PDF
【メタサーベイ】基盤モデル / Foundation Models
bycvpaper. challenge
 
PPTX
[DL輪読会]Neural Ordinary Differential Equations
byDeep Learning JP
 
PDF
Active Learning 入門
byShuyo Nakatani
 
PPTX
[DL輪読会]ドメイン転移と不変表現に関するサーベイ
byDeep Learning JP
 
PPTX
【DL輪読会】言語以外でのTransformerのまとめ (ViT, Perceiver, Frozen Pretrained Transformer etc)
byDeep Learning JP
 
PDF
semantic segmentation サーベイ
byyohei okawa
 
Graph Neural Networks
bytm1966
 
ドメイン適応の原理と応用
byYoshitaka Ushiku
 
【メタサーベイ】基盤モデル / Foundation Models
bycvpaper. challenge
 
[DL輪読会]Neural Ordinary Differential Equations
byDeep Learning JP
 
Active Learning 入門
byShuyo Nakatani
 
[DL輪読会]ドメイン転移と不変表現に関するサーベイ
byDeep Learning JP
 
【DL輪読会】言語以外でのTransformerのまとめ (ViT, Perceiver, Frozen Pretrained Transformer etc)
byDeep Learning JP
 
semantic segmentation サーベイ
byyohei okawa
 

What's hot

PDF
CVIM#11 3. 最小化のための数値計算
bysleepy_yoshi
 
PDF
機械学習による統計的実験計画（ベイズ最適化を中心に）
byKota Matsui
 
PDF
目で見る過学習と正則化
byy-uti
 
PDF
Optimizer入門＆最新動向
byMotokawa Tetsuya
 
PDF
【メタサーベイ】数式ドリブン教師あり学習
bycvpaper. challenge
 
PPTX
Curriculum Learning （関東CV勉強会）
byYoshitaka Ushiku
 
PPTX
【DL輪読会】The Forward-Forward Algorithm: Some Preliminary
byDeep Learning JP
 
PDF
最近のDeep Learning (NLP) 界隈におけるAttention事情
byYuta Kikuchi
 
PDF
【メタサーベイ】Video Transformer
bycvpaper. challenge
 
PDF
[DL輪読会]Relational inductive biases, deep learning, and graph networks
byDeep Learning JP
 
PPTX
【LT資料】 Neural Network 素人なんだけど何とかご機嫌取りをしたい
byTakuji Tahara
 
PDF
Transformer 動向調査 in 画像認識
byKazuki Maeno
 
PDF
20190619 オートエンコーダーと異常検知入門
byKazuki Motohashi
 
PDF
[DL輪読会]近年のエネルギーベースモデルの進展
byDeep Learning JP
 
PPTX
[DL輪読会]Revisiting Deep Learning Models for Tabular Data (NeurIPS 2021) 表形式デー...
byDeep Learning JP
 
PPTX
[DL輪読会]Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets
byDeep Learning JP
 
PDF
[DL輪読会]NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis
byDeep Learning JP
 
PDF
ブラックボックス最適化とその応用
bygree_tech
 
PDF
AutoEncoderで特徴抽出
byKai Sasaki
 
PDF
工学系大学4年生のための論文の読み方
byychtanaka
 
CVIM#11 3. 最小化のための数値計算
bysleepy_yoshi
 
機械学習による統計的実験計画（ベイズ最適化を中心に）
byKota Matsui
 
目で見る過学習と正則化
byy-uti
 
Optimizer入門＆最新動向
byMotokawa Tetsuya
 
【メタサーベイ】数式ドリブン教師あり学習
bycvpaper. challenge
 
Curriculum Learning （関東CV勉強会）
byYoshitaka Ushiku
 
【DL輪読会】The Forward-Forward Algorithm: Some Preliminary
byDeep Learning JP
 
最近のDeep Learning (NLP) 界隈におけるAttention事情
byYuta Kikuchi
 
【メタサーベイ】Video Transformer
bycvpaper. challenge
 
[DL輪読会]Relational inductive biases, deep learning, and graph networks
byDeep Learning JP
 
【LT資料】 Neural Network 素人なんだけど何とかご機嫌取りをしたい
byTakuji Tahara
 
Transformer 動向調査 in 画像認識
byKazuki Maeno
 
20190619 オートエンコーダーと異常検知入門
byKazuki Motohashi
 
[DL輪読会]近年のエネルギーベースモデルの進展
byDeep Learning JP
 
[DL輪読会]Revisiting Deep Learning Models for Tabular Data (NeurIPS 2021) 表形式デー...
byDeep Learning JP
 
[DL輪読会]Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets
byDeep Learning JP
 
[DL輪読会]NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis
byDeep Learning JP
 
ブラックボックス最適化とその応用
bygree_tech
 
AutoEncoderで特徴抽出
byKai Sasaki
 
工学系大学4年生のための論文の読み方
byychtanaka
 

Similar to Hyperbolic Neural Networks

PDF
Hyperbolic Deep Reinforcement Learning
bySangwoo Mo
 
PPTX
Poincare embeddings for Learning Hierarchical Representations
byTatsuya Shirakawa
 
PPTX
Hyperbolic Image Embedding.pptx
bytaeseon ryu
 
PDF
[ICLR/ICML2019読み会] A Wrapped Normal Distribution on Hyperbolic Space for Grad...
byYoshihiro Nagano
 
PPTX
Fassold-MMAsia2023-Tutorial-GeometricDL-Part1.pptx
byHannesFesswald
 
PPTX
20180831 riemannian representation learning
bysegwangkim
 
Hyperbolic Deep Reinforcement Learning
bySangwoo Mo
 
Poincare embeddings for Learning Hierarchical Representations
byTatsuya Shirakawa
 
Hyperbolic Image Embedding.pptx
bytaeseon ryu
 
[ICLR/ICML2019読み会] A Wrapped Normal Distribution on Hyperbolic Space for Grad...
byYoshihiro Nagano
 
Fassold-MMAsia2023-Tutorial-GeometricDL-Part1.pptx
byHannesFesswald
 
20180831 riemannian representation learning
bysegwangkim
 

More from Tatsuya Shirakawa

PDF
データに内在する構造をみるための埋め込み手法
byTatsuya Shirakawa
 
PDF
2021 10-07 kdd2021読み会 uc phrase
byTatsuya Shirakawa
 
PDF
Improving Variational Inference with Inverse Autoregressive Flow
byTatsuya Shirakawa
 
PDF
Retail Face Analysis Inside-Out
byTatsuya Shirakawa
 
PDF
Seeing Unseens with Machine Learning -- 
見えていないものを見出す機械学習
byTatsuya Shirakawa
 
PDF
Icml2017 overview
byTatsuya Shirakawa
 
PDF
ICCV2019 report
byTatsuya Shirakawa
 
PDF
ヒトの機械学習
byTatsuya Shirakawa
 
PDF
NeurIPS2021読み会 Fairness in Ranking under Uncertainty
byTatsuya Shirakawa
 
PDF
Learning to Compose Domain-Specific Transformations for Data Augmentation
byTatsuya Shirakawa
 
PDF
Dynamic filter networks
byTatsuya Shirakawa
 
PDF
Taskonomy: Disentangling Task Transfer Learning -- Scouty Meetup 2018 Feb., ...
byTatsuya Shirakawa
 
PDF
医療ビッグデータの今後を見通すために知っておきたい機械学習の基礎〜最前線 agains COVID-19
byTatsuya Shirakawa
 
データに内在する構造をみるための埋め込み手法
byTatsuya Shirakawa
 
2021 10-07 kdd2021読み会 uc phrase
byTatsuya Shirakawa
 
Improving Variational Inference with Inverse Autoregressive Flow
byTatsuya Shirakawa
 
Retail Face Analysis Inside-Out
byTatsuya Shirakawa
 
Seeing Unseens with Machine Learning -- 
見えていないものを見出す機械学習
byTatsuya Shirakawa
 
Icml2017 overview
byTatsuya Shirakawa
 
ICCV2019 report
byTatsuya Shirakawa
 
ヒトの機械学習
byTatsuya Shirakawa
 
NeurIPS2021読み会 Fairness in Ranking under Uncertainty
byTatsuya Shirakawa
 
Learning to Compose Domain-Specific Transformations for Data Augmentation
byTatsuya Shirakawa
 
Dynamic filter networks
byTatsuya Shirakawa
 
Taskonomy: Disentangling Task Transfer Learning -- Scouty Meetup 2018 Feb., ...
byTatsuya Shirakawa
 
医療ビッグデータの今後を見通すために知っておきたい機械学習の基礎〜最前線 agains COVID-19
byTatsuya Shirakawa
 

Recently uploaded

PDF
Running Non-Cloud-Native Databases in Cloud-Native Environments_ Challenges a...
byAlkin Tezuysal
 
PDF
KMWorld - KM & AI Bring Collectivity, Nostalgia, & Selectivity
byJonathan Ralton
 
PDF
Mastering UiPath Maestro – Session 2 – Building a Live Use Case - Session 2
byDianaGray10
 
PDF
Build an AI/ML-driven image archive processing workflow: Image archive, analy...
bywesley chun
 
PDF
Dev Dives: Build smarter agents with UiPath Agent Builder
byUiPathCommunity
 
PPTX
oop pillor polymorphism Code Presentation
byhifzamahmood8
 
PDF
Beyond Basics: How to Build Scalable, Intelligent Imagery Pipelines
bySafe Software
 
PPTX
Explaining ourselves – people, computers and AI
byAlan Dix
 
PDF
OutSystsems User Group Brabant November 2025
bymail496323
 
PPTX
Support, Monitoring, Continuous Improvement & Scaling Agentic Automation [3/3]
byUiPathCommunity
 
PDF
UnityNet AI Induced Harm & Youth Protection Position Statement 2025-11-15
byLHelferty
 
PDF
Mutation Testing for Industrial Robotic Systems (FMAS 2025)
bySylvain Hallé
 
PDF
PMI PMBOK8 Released on November 13, 2025
byScott M. Graffius
 
PDF
UiPath DevConnect 2025: UiPath ScreenPlay - The Future of Human-Like Automation
byDianaGray10
 
PDF
IDSECCONF2025 - Faisal Ilham - Semi Automating Vulnerability Scanner and Expl...
byidsecconf
 
PPTX
Streamlining Business Processes with UiPath Apps
byTracy Dixon
 
PPTX
Neo4j Fraud GraphTalk Singapore Nov 2025
bygourisachdeva2
 
PDF
LVM Management and Disaster Recovery.pdf
byLinuxCert Guru
 
PDF
Linux Foundation Certified System Administrator (LFCS) Exam.pdf
byLinuxCert Guru
 
PDF
SELinux Basics: Managing SELinux Modes and Context - RHCSA+.pdf
byLinuxCert Guru
 
Running Non-Cloud-Native Databases in Cloud-Native Environments_ Challenges a...
byAlkin Tezuysal
 
KMWorld - KM & AI Bring Collectivity, Nostalgia, & Selectivity
byJonathan Ralton
 
Mastering UiPath Maestro – Session 2 – Building a Live Use Case - Session 2
byDianaGray10
 
Build an AI/ML-driven image archive processing workflow: Image archive, analy...
bywesley chun
 
Dev Dives: Build smarter agents with UiPath Agent Builder
byUiPathCommunity
 
oop pillor polymorphism Code Presentation
byhifzamahmood8
 
Beyond Basics: How to Build Scalable, Intelligent Imagery Pipelines
bySafe Software
 
Explaining ourselves – people, computers and AI
byAlan Dix
 
OutSystsems User Group Brabant November 2025
bymail496323
 
Support, Monitoring, Continuous Improvement & Scaling Agentic Automation [3/3]
byUiPathCommunity
 
UnityNet AI Induced Harm & Youth Protection Position Statement 2025-11-15
byLHelferty
 
Mutation Testing for Industrial Robotic Systems (FMAS 2025)
bySylvain Hallé
 
PMI PMBOK8 Released on November 13, 2025
byScott M. Graffius
 
UiPath DevConnect 2025: UiPath ScreenPlay - The Future of Human-Like Automation
byDianaGray10
 
IDSECCONF2025 - Faisal Ilham - Semi Automating Vulnerability Scanner and Expl...
byidsecconf
 
Streamlining Business Processes with UiPath Apps
byTracy Dixon
 
Neo4j Fraud GraphTalk Singapore Nov 2025
bygourisachdeva2
 
LVM Management and Disaster Recovery.pdf
byLinuxCert Guru
 
Linux Foundation Certified System Administrator (LFCS) Exam.pdf
byLinuxCert Guru
 
SELinux Basics: Managing SELinux Modes and Context - RHCSA+.pdf
byLinuxCert Guru
 

Hyperbolic Neural Networks

  • 1.
    Hyperbolic Neural Networks 2019Jan. 26th Tatsuya Shirakawa (ABEJA, Inc.)
  • 2.
    ABEJA, Inc. (Researcher) -Deep Learning (CV, Graph, NLP, ) - Machine Learning - Mathematical Optimization - https://github.com/TatsuyaShirakawa Tech blog http://tech-blog.abeja.asia/ Poincaré Embeddings Graph Convolution Annotation Self Intro. Hyperbolic
  • 3.
    This paper From myblog article
  • 4.
    Paper Intro. O. E.Ganea+, “Hyperbolic Neural Networks”, NeurIPS2018 (spotlight) • Exploring the connection between 
 Möbius gyrovector space and Riemannian hyperbolic geometry • Construction of several components of neural networks in hyperbolic space • Experiments which prove the efficiency of the proposed construction
  • 5.
    Methods in HyperbolicSpaces [M. Nickel+]
 Poincaré Embeddings ~ 17’ 05 18’ 04 [C. D. Sa+]
 Representation Tradeoff ~
 (Near-exact tree embs., h-MDS) [O. Ganea+]
 Hyperbolic Entailment Cones
 (Poincare embs. + Order embs.) 18’ 05 [C. Gulcehre+]
 Hyperbolic Attention Networks
 (Einstein Mid. Point) [O. Ganea+]
 Hyperbolic Neural Networks [M. Nickel+]
 Learning Continuous Hierarchies
 in the Lorentz Model~ [A. Tifrea+]
 Poincaré Glove ~ (Poincaré Glove) 18’ 06 18’ 10 …
  • 6.
    Embeddings [M. Nickel+]
 Poincaré Embeddings~ 17’ 05 18’ 04 [C. D. Sa+]
 Representation Tradeoff ~
 (Near-exact tree embs., h-MDS) [O. Ganea+]
 Hyperbolic Entailment Cones
 (Poincare embs. + Order embs.) 18’ 05 [C. Gulcehre+]
 Hyperbolic Attention Networks
 (Einstein Mid. Point) [O. Ganea+]
 Hyperbolic Neural Networks [M. Nickel+]
 Learning Continuous Hierarchies
 in the Lorentz Model~ [A. Tifrea+]
 Poincaré Glove ~ (Poincaré Glove) 18’ 06 18’ 10 …
  • 7.
    Neural Network Construction [M.Nickel+]
 Poincaré Embeddings ~ 17’ 05 18’ 04 [C. D. Sa+]
 Representation Tradeoff ~
 (Near-exact tree embs., h-MDS) [O. Ganea+]
 Hyperbolic Entailment Cones
 (Poincare embs. + Order embs.) 18’ 05 [C. Gulcehre+]
 Hyperbolic Attention Networks
 (Einstein Mid. Point) [O. Ganea+]
 Hyperbolic Neural Networks [M. Nickel+]
 Learning Continuous Hierarchies
 in the Lorentz Model~ [A. Tifrea+]
 Poincaré Glove ~ (Poincaré Glove) 18’ 06 18’ 10 …
  • 8.
    Hyperbolic Space • Manifoldswith constant sectional curvature
 - zero curvature: Euclidean space
 - pos. curvature: hypersphare
 - neg. curvature: hyperbolic space • Equivalent models of hyperbolic space
 - the upper half-space model
 - the poincaré disk model
 - the Beltrami-Klein model
 - the Lorentz model (hyperboloid model)
 …
  • 9.
    Poincaré Disk/Ball Model •A model for hyperbolic space, which is suitable for visualization • A unit-Ball equipped with the following Riemannian metric • hyperbolic 
 The volume of a ball grows 
 exponentially with the radius
 tree 
 the number of nodes grows 
 exponentially with the depth 「異空間散歩！双曲空間を歩いてみよう。」
 (http://nunuki.hatenablog.com/entry/2018/12/15/055136)
  • 10.
    Euclidean vs Hyperbolic EuclideanHyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Gyrovector Space
  • 11.
    Euclidean vs Hyperbolic EuclideanHyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b Möbius transform, relection Algebraic structure Vector Space Gyrovector Space
  • 12.
    Euclidean vs Hyperbolic EuclideanHyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Gyrovector Space M¨obius transformation in 2d (with polar decomp.) az + b cz + d = ei✓ w(z), w(z) = z + w ¯wz + 1 = · · · = (1 + 2w · z + kzk2 )w + (1 kw2 k2 )z 1 + 2w · z + kwk2kzk2 <latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit><latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit><latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit><latexit sha1_base64="kVqp236jgr2jvlciZjGbsx+UgDA=">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</latexit>
  • 13.
    Euclidean vs Hyperbolic EuclideanHyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Gyrovector Space z M¨obius transformation a(x) = (1 + 2a · x + kxk2 )a + (1 kak2 )x 1 + 2a · x + kak2kxk2 or, with scale parameter c > 0, c a(x) = (1 + 2ca · x + ckxk2 )a + (1 ckak2 )x 1 + 2ca · x + c2kak2kxk2 <latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">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</latexit><latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">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</latexit><latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">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</latexit><latexit sha1_base64="8KLwFiPSQ43Z4+hsT/i4J3DO7Go=">AAAEOXiclVNNb9NAEN3afBTz0RSOXEaNqFK1RHaEBBdQBRcuiFZq0krZJFpv1s2qa6+1u4ZGlv8WF/4FNyQuHECIK3+AjWvkNimEjrTS05s3O7NPs2EquDa+/3nFca9dv3Fz9ZZ3+87de2uN9fs9LTNFWZdKIdVRSDQTPGFdw41gR6liJA4FOwxPXs3yh++Y0lwmB2aaskFMjhMecUqMpUbrzttNbNipyd/gDRnyTINRJNGRVHGpKABjbxOnEz7KsaUmYZSTomj9wafFFjwHHClCc68VwDZ0oNYBpmNpoBbbPO4xdYEqiWFn63zdNti7Hs9pSa2ty70i967etr7qL/PAsFM9vXRHqh14z80ENCWCQUoUiZlhCgqg8AL8ncttGtJ/GkWXzEyvZNa8eoldS5svePNfpo0aTb/tlwGLIKhAE1WxN2p8wmNJs5glhgqidT/wUzPIiTKcClZ4ONMsJfSEHLO+hYk1Xg/ycvMLeGSZMdhttScxULLnK3ISaz2NQ6uczarnczPyslw/M9GzQc6TNDMsoWeNokyAkTD7RjDmilEjphYQqridFejEbgW1S6E9a0Iw/+RF0Ou0A78d7D9p7r6s7FhFD9EGaqEAPUW76DXaQ11EnQ/OF+eb89396H51f7g/z6TOSlXzAF0I99dvbkVd3w==</latexit>
  • 14.
    Euclidean vs Hyperbolic EuclideanHyperblic (Poincaré) Metric Flat Geodesic
 (shortest curve) Line Line / Circle Isometry Ax+b (A: orthogonal) Möbius transform, relection Algebraic structure Vector Space Möbius Gyrovector Space vector M¨obius gyrovector x 7! x + a x 7! a x := a(x) = a x(0) x 7! rx x 7! r ⌦ x (continuation of n x = x x · · · x)<latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit>
  • 15.
    More Gyrovector space •Gyrovector space can be also defined for other hyperbolic models
 - Einstein gyrovector space <—> algebraic formulation of special relativity • Relation to Riemannian hyperbolic geometry
 - • Mid point • Gyrotrigonometry
 … a ( a b) ⌦ t ! geodesic from a to b<latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit><latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">AAACd3icbZHLThsxFIY9Ay0QegmwgwUWoVW6iWYqJFhGsOmSSg0gZaLI45yZWPgyss8A0WhegYdjx3t00109IUhAOJKlX99/bJ9LWkjhMIoeg3Bl9cPHtfWN1uanz1++tre2L5wpLYcBN9LYq5Q5kELDAAVKuCosMJVKuEyvzxr/8gasE0b/wVkBI8VyLTLBGXo0bt8niuE0zSpW08QUsnS0mxgltBfL1jNJ6x8eoVDgKNJEGp1LyNCKfIrMWnNLE4Q7rHIwE3CC08waRetXL84TKJoXOK3H7U7Ui+ZBl0W8EB2yiPNx+yGZGF4q0Mglc24YRwWOKmZRcAl1KykdFIxfsxyGXmrmSx5V87nV9JsnE5oZ649GOqcvb1RMOTdTqc9sKnRvvQa+5w1LzE5GldBFiaD500dZKZtumyXQibDAUc68YNwKXyvlU2YZR7+qlh9C/LblZXHxsxdHvfj3Uad/uhjHOtkjB6RLYnJM+uQXOScDwsnfYDfoBIfBv3A//B52n1LDYHFnh7yKMP4PNSTB1g==</latexit><latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit><latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit>
  • 16.
    Riemannian Hyperbolic Geometry
 exp/logmap • exp: map from tangent space to manifold • log: reversed map from manifold to tangent space • In hyperbolic space, exp/log map 
 extend to whole space, having 
 closed form of
  • 17.
    Hyperbolic Neural Networks EuclideanHyperbolic Softmax —
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit>
  • 18.
    Hyperbolic Neural Networks EuclideanHyperbolic Softmax —
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">AAACYnicbVFNa+MwFJTdj816+5E0x92D2LCQUgh2KLSXQuhe9lLowiYpxMHIynMiKstGel5qTP5kbz310h+ycupDtu0DwWjePI00inMpDPr+k+Pu7O7tf2p99r4cHB4dtzsnE5MVmsOYZzLTdzEzIIWCMQqUcJdrYGksYRrf/6z707+gjcjUHyxzmKdsqUQiOENLRe2yfxMFIb2Jhqc0lJBg3wtjWApVMa1Zua6kXHthynAVJ9XD2mrD7e3QC0EtGq0XarFc4Sm9sucFdHuKntUW29Qwavf8gb8p+h4EDeiRpm6j9mO4yHiRgkIumTGzwM9xbr1RcAnWvTCQM37PljCzULEUzLzaRLSmPyyzoEmm7VJIN+z2RMVSY8o0tsr6juZtryY/6s0KTC7nlVB5gaD4q1FSSIoZrfOmC6GBoywtYFwLe1fKV0wzjvZXPBtC8PbJ78FkOAj8QfD7vDe6buJoka/kO+mTgFyQEflFbsmYcPLs7DlHzrHz4npux+2+Sl2nmemS/8r99g+OAbTf</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">AAACYnicbVFNa+MwFJTdj816+5E0x92D2LCQUgh2KLSXQuhe9lLowiYpxMHIynMiKstGel5qTP5kbz310h+ycupDtu0DwWjePI00inMpDPr+k+Pu7O7tf2p99r4cHB4dtzsnE5MVmsOYZzLTdzEzIIWCMQqUcJdrYGksYRrf/6z707+gjcjUHyxzmKdsqUQiOENLRe2yfxMFIb2Jhqc0lJBg3wtjWApVMa1Zua6kXHthynAVJ9XD2mrD7e3QC0EtGq0XarFc4Sm9sucFdHuKntUW29Qwavf8gb8p+h4EDeiRpm6j9mO4yHiRgkIumTGzwM9xbr1RcAnWvTCQM37PljCzULEUzLzaRLSmPyyzoEmm7VJIN+z2RMVSY8o0tsr6juZtryY/6s0KTC7nlVB5gaD4q1FSSIoZrfOmC6GBoywtYFwLe1fKV0wzjvZXPBtC8PbJ78FkOAj8QfD7vDe6buJoka/kO+mTgFyQEflFbsmYcPLs7DlHzrHz4npux+2+Sl2nmemS/8r99g+OAbTf</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit> “Hyperplane margin classifiers on the multinomial manifold”, NIPS2014 Euclidean ak · x bk = ak · (x pk) (ifbk = ak · pk) = kakksign(ak · (x pk))d(x, {y|ak · (y pk) = 0}) Hyperbolic q gp(ak, ak)sign(ak · (x pk))d(x, {y|ak · (y pk) = 0}) <latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">AAAEDXicrVPPi9QwFM62/ljHHzurRy/BwWUG1qEVwb0sLIqwxxWc2YXJUNI0nQ2TJjVJZUvtP+DFf8WLB0W8evfmf2Pa7eh0ZhQEHxRevve9vPd9NGHKmTae92PLca9cvXZ9+0bn5q3bd3a6u3fHWmaK0BGRXKqzEGvKmaAjwwynZ6miOAk5PQ3nz6v66RuqNJPilclTOk3wTLCYEWwsFOw6vT1k6IUpXmSEs4hiUUKEOnsoweY8jAtcBnOISCQNXEAXJXwEwwq2vEO4gdlvUReH1BIGCPZZXHdv7GxxFwPGVJkVcg1pNhP9fxs/gNFSdR+i4lc9L+Hbv4rJ126zGjxU1nvWntVGHluXVSg5I42T+rUyxSz43dnaeb81c1D+SdWyKCQTJjL9X6VtvLMRGHR73tCrA64nfpP0QBMnQfc7iiTJEioM4Vjrie+lZlpgZRjhtOygTNMUkzme0YlNBU6onhb131zChxaJYCyV/YSBNbrcUeBE6zwJLbPaVa/WKnBTbZKZ+GBaMJFmhgpyOSjOODQSVk8DRkxRYnhuE0wUs7tCco4VJsY+oI41wV+VvJ6MHw99b+i/fNI7etbYsQ3ugwegD3zwFByBY3ACRoA475wPzifns/ve/eh+cb9eUp2tpuceaIX77ScXkVSi</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="hP+6LrUf2d3tZaldqaQQvEKMXyw=">AAAB2XicbZDNSgMxFIXv1L86Vq1rN8EiuCozbnQpuHFZwbZCO5RM5k4bmskMyR2hDH0BF25EfC93vo3pz0JbDwQ+zknIvSculLQUBN9ebWd3b/+gfugfNfzjk9Nmo2fz0gjsilzl5jnmFpXU2CVJCp8LgzyLFfbj6f0i77+gsTLXTzQrMMr4WMtUCk7O6oyaraAdLMW2IVxDC9YaNb+GSS7KDDUJxa0dhEFBUcUNSaFw7g9LiwUXUz7GgUPNM7RRtRxzzi6dk7A0N+5oYkv394uKZ9bOstjdzDhN7Ga2MP/LBiWlt1EldVESarH6KC0Vo5wtdmaJNChIzRxwYaSblYkJN1yQa8Z3HYSbG29D77odBu3wMYA6nMMFXEEIN3AHD9CBLghI4BXevYn35n2suqp569LO4I+8zx84xIo4</latexit><latexit sha1_base64="mXej7eo96vAzELZ/uggt3i9doxs=">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</latexit><latexit sha1_base64="mXej7eo96vAzELZ/uggt3i9doxs=">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</latexit><latexit sha1_base64="emtCgJQUpEekBRkSlrXZJOYv+Cc=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">AAAEDXicrVPPi9QwFM62/ljHHzurRy/BwWUG1qEVwb0sLIqwxxWc2YXJUNI0nQ2TJjVJZUvtP+DFf8WLB0W8evfmf2Pa7eh0ZhQEHxRevve9vPd9NGHKmTae92PLca9cvXZ9+0bn5q3bd3a6u3fHWmaK0BGRXKqzEGvKmaAjwwynZ6miOAk5PQ3nz6v66RuqNJPilclTOk3wTLCYEWwsFOw6vT1k6IUpXmSEs4hiUUKEOnsoweY8jAtcBnOISCQNXEAXJXwEwwq2vEO4gdlvUReH1BIGCPZZXHdv7GxxFwPGVJkVcg1pNhP9fxs/gNFSdR+i4lc9L+Hbv4rJ126zGjxU1nvWntVGHluXVSg5I42T+rUyxSz43dnaeb81c1D+SdWyKCQTJjL9X6VtvLMRGHR73tCrA64nfpP0QBMnQfc7iiTJEioM4Vjrie+lZlpgZRjhtOygTNMUkzme0YlNBU6onhb131zChxaJYCyV/YSBNbrcUeBE6zwJLbPaVa/WKnBTbZKZ+GBaMJFmhgpyOSjOODQSVk8DRkxRYnhuE0wUs7tCco4VJsY+oI41wV+VvJ6MHw99b+i/fNI7etbYsQ3ugwegD3zwFByBY3ACRoA475wPzifns/ve/eh+cb9eUp2tpuceaIX77ScXkVSi</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">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</latexit><latexit sha1_base64="nXAo5BEtaflaTA5Rtjw2LYjntjM=">AAAEDXicrVPPi9QwFM62/ljHHzurRy/BwWUG1qEVwb0sLIqwxxWc2YXJUNI0nQ2TJjVJZUvtP+DFf8WLB0W8evfmf2Pa7eh0ZhQEHxRevve9vPd9NGHKmTae92PLca9cvXZ9+0bn5q3bd3a6u3fHWmaK0BGRXKqzEGvKmaAjwwynZ6miOAk5PQ3nz6v66RuqNJPilclTOk3wTLCYEWwsFOw6vT1k6IUpXmSEs4hiUUKEOnsoweY8jAtcBnOISCQNXEAXJXwEwwq2vEO4gdlvUReH1BIGCPZZXHdv7GxxFwPGVJkVcg1pNhP9fxs/gNFSdR+i4lc9L+Hbv4rJ126zGjxU1nvWntVGHluXVSg5I42T+rUyxSz43dnaeb81c1D+SdWyKCQTJjL9X6VtvLMRGHR73tCrA64nfpP0QBMnQfc7iiTJEioM4Vjrie+lZlpgZRjhtOygTNMUkzme0YlNBU6onhb131zChxaJYCyV/YSBNbrcUeBE6zwJLbPaVa/WKnBTbZKZ+GBaMJFmhgpyOSjOODQSVk8DRkxRYnhuE0wUs7tCco4VJsY+oI41wV+VvJ6MHw99b+i/fNI7etbYsQ3ugwegD3zwFByBY3ACRoA475wPzifns/ve/eh+cb9eUp2tpuceaIX77ScXkVSi</latexit>
  • 19.
    Hyperbolic Neural Networks EuclideanHyperbolic Softmax —
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit>
  • 20.
    Hyperbolic Neural Networks EuclideanHyperbolic Softmax — 
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit> extend
  • 21.
    Hyperbolic Neural Networks EuclideanHyperbolic Softmax — 
 Multinomial Logistic Regression (MLR) Scalar Multiplication Applying Function
 - linear transform
 - activation function
 - elementwise mult Matrix mult. 
 after concat rx<latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit><latexit sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit><latexit sha1_base64="AHaYGenMiw70KcZ5eluaVacsPqI=">AAACCHicbZC7TsMwFIYdrqXcAowMWFRI7VIlCAnGChbGItGL1ESR4zqtVceObAe1ijqy8CosDCDEyiOw8TY4bQZo+SVLn/5zjnzOHyaMKu0439bK6tr6xmZpq7y9s7u3bx8ctpVIJSYtLJiQ3RApwignLU01I91EEhSHjHTC0U1e7zwQqajg93qSED9GA04jipE2VmCfeGScBA6sRlWPiUFOXoz0MIyy8bRWqwV2xak7M8FlcAuogELNwP7y+gKnMeEaM6RUz3US7WdIaooZmZa9VJEE4REakJ5BjmKi/Gx2yBSeGacPIyHN4xrO3N8TGYqVmsSh6cyXVIu13Pyv1kt1dOVnlCepJhzPP4pSBrWAeSqwTyXBmk0MICyp2RXiIZIIa5Nd2YTgLp68DO3zuuvU3buLSuO6iKMEjsEpqAIXXIIGuAVN0AIYPIJn8ArerCfrxXq3PuatK1YxcwT+yPr8AR4zmB0=</latexit> (M1 M2) ✓ x1 x2 ◆ = M1x1 + M2x2 <latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">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</latexit><latexit sha1_base64="pYZY7DvlQq586lwO4H04718zVk8=">AAACYnicbVFNa+MwFJTdj816+5E0x92D2LCQUgh2KLSXQuhe9lLowiYpxMHIynMiKstGel5qTP5kbz310h+ycupDtu0DwWjePI00inMpDPr+k+Pu7O7tf2p99r4cHB4dtzsnE5MVmsOYZzLTdzEzIIWCMQqUcJdrYGksYRrf/6z707+gjcjUHyxzmKdsqUQiOENLRe2yfxMFIb2Jhqc0lJBg3wtjWApVMa1Zua6kXHthynAVJ9XD2mrD7e3QC0EtGq0XarFc4Sm9sucFdHuKntUW29Qwavf8gb8p+h4EDeiRpm6j9mO4yHiRgkIumTGzwM9xbr1RcAnWvTCQM37PljCzULEUzLzaRLSmPyyzoEmm7VJIN+z2RMVSY8o0tsr6juZtryY/6s0KTC7nlVB5gaD4q1FSSIoZrfOmC6GBoywtYFwLe1fKV0wzjvZXPBtC8PbJ78FkOAj8QfD7vDe6buJoka/kO+mTgFyQEflFbsmYcPLs7DlHzrHz4npux+2+Sl2nmemS/8r99g+OAbTf</latexit> M1 ⌦ x1 M2 ⌦ x2<latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit><latexit sha1_base64="hCyRWRaGVdOUSH6Wre83xTrLEFc=">AAACIXicbVDLSsNAFJ34rPUVdelmsAiuSlIEuyy6cVOoYB/QhjCZTtqhk5kwMxFL6K+48VfcuFCkO/FnnKRZaNsDFw7n3Mu99wQxo0o7zre1sbm1vbNb2ivvHxweHdsnpx0lEolJGwsmZC9AijDKSVtTzUgvlgRFASPdYHKX+d0nIhUV/FFPY+JFaMRpSDHSRvLtetN34UBoGhEFBxHS4yBMn2e5GLNEwaZfW+fXfLviVJ0ccJW4BamAAi3fng+GAicR4RozpFTfdWLtpUhqihmZlQeJIjHCEzQifUM5Mhu9NP9wBi+NMoShkKa4hrn6dyJFkVLTKDCd2Y1q2cvEdV4/0WHdSymPE004XiwKEwa1gFlccEglwZpNDUFYUnMrxGMkEdYm1LIJwV1+eZV0alXXqboP15XGbRFHCZyDC3AFXHADGuAetEAbYPAC3sAH+LRerXfry5ovWjesYuYM/IP18wvlM6Nc</latexit>
  • 22.
    Hyperbolic RNN/GRU • HyperbolicRNN • Hyperbolic GRU

  • 23.
  • 24.
    Task1 — SNLI •SNLI (Stanford Natural Language Inference) — textual entailment
 
 - sentence A: “Little kids A. And B. Are playing soccer.”
 - sentence B: “Two children are playing outdoors.”
 
 Q) A => B?
  • 25.
    Task2 — PREFIX-Z% •PREFIX-Z% (Z=10, 30, 50) — PoC task
 
 - sent1 (length=n1) = random seq. of symbols (words)
 - sent2 (length=n2) = random seq. of symbols (len(sent2) <= len(sent1))
 - sent1’(length=n2) = take first n2 symbols of sent1 and replace Z% symbols with randomized ones
 
 Q) detect whether (s1, s2) comes from (sent1, sent2) or (sent1, sent1’)
 (s1, s2) Label s1 = 23, 42, 52, 29, 20, 22, 22, 11, …, 60, 16
 s2 = 58, 17, 100, 27, 10, 37, 24, 50 0 s1 = 23, 42, 52, 29, 20, 22, 22, 11, …, 60, 16
 s2 = 66, 42, 52, 29, 20, 22, 38, 11 1
  • 26.
  • 27.
    Decision Boundary (SNLI) •No advantage for hyperbolic MLR in SNLI task
  • 28.
    MLR classification experiments PoincaréEmbs.
 (https://github.com/facebookresearch/poincare-embeddings) Take a subtree
 A = node in the subtree
 B = node not in the subtree Classify with - hyperbolic MLR - Euclidean MLR - log_0 + Euclidean MLR
  • 29.
    More Details andRelated Works https://tech-blog.abeja.asia/entry/hyperbolic_ml_2019