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Hyperbolic Neural Networks

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slides describing the following paper:

Octavian-Eugen Ganea et. al., "Hyperbolic Neural Networks", NeurIPS 2018

(NeurIPS 2018 paper reading time at PFN at 2019 Jan. 26th)

Published in: Technology
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Hyperbolic Neural Networks

  1. 1. Hyperbolic Neural Networks 2019 Jan. 26th Tatsuya Shirakawa (ABEJA, Inc.)
  2. 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. 3. This paper From my blog article
  4. 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. 5. 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 …
  6. 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. 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. 8. 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)
 …
  9. 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. 10. 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
  11. 11. 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
  12. 12. 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>
  13. 13. 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=">AAAEOXiclVNNb9NAEN3afBTz0RSOXEaNqFK1RHaEBBdQBRcuiFZq0krZJFpv1s2qa6+1u4ZGlv8WF/4FNyQuHECIK3+AjWvkNimEjrTS05s3O7NPs2EquDa+/3nFca9dv3Fz9ZZ3+87de2uN9fs9LTNFWZdKIdVRSDQTPGFdw41gR6liJA4FOwxPXs3yh++Y0lwmB2aaskFMjhMecUqMpUbrzttNbNipyd/gDRnyTINRJNGRVHGpKABjbxOnEz7KsaUmYZSTomj9wafFFjwHHClCc68VwDZ0oNYBpmNpoBbbPO4xdYEqiWFn63zdNti7Hs9pSa2ty70i967etr7qL/PAsFM9vXRHqh14z80ENCWCQUoUiZlhCgqg8AL8ncttGtJ/GkWXzEyvZNa8eoldS5svePNfpo0aTb/tlwGLIKhAE1WxN2p8wmNJs5glhgqidT/wUzPIiTKcClZ4ONMsJfSEHLO+hYk1Xg/ycvMLeGSZMdhttScxULLnK3ISaz2NQ6uczarnczPyslw/M9GzQc6TNDMsoWeNokyAkTD7RjDmilEjphYQqridFejEbgW1S6E9a0Iw/+RF0Ou0A78d7D9p7r6s7FhFD9EGaqEAPUW76DXaQ11EnQ/OF+eb89396H51f7g/z6TOSlXzAF0I99dvbkVd3w==</latexit>
  14. 14. 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=">AAADvHicfVJba9swFFbsXbrslm6PexENGymDYJfB2kFH2V72MuhgaQtRCLIsJyKyZKTjEGPyJ/u2fzPZcZukyXZAcM73nbtOlElhIQj+tDz/0eMnTw+etZ+/ePnqdefwzZXVuWF8wLTU5iailkuh+AAESH6TGU7TSPLraPa94q/n3Fih1W8oMj5K6USJRDAKDhoftm5JxCdCldQYWixLKZdtAnwB5Zwz0GaJP+CV/ZMc6UjkFk8Ko+9IQtokpTCNknLhrJRmFjTegD7eG7RO9W9nxxOdSVdgw+nLOSbZVIzXTr01e4x3WEyYMGwbXaxjAhezv2WzWfW/fVZTaxAp32qUNFvqMa1AqLxeL9bJktx7qX0Dnm8l2aHvEBZrsPv4uuix+zOu4uYP2+NON+gHteBdJWyULmrkcty5JbFmecoVMEmtHYZBBiOXDgST3CXPLc8om9EJHzpVUTf8qKyPb4nfOyTGiTbuKcA1uhlR0tTaIo2cZ9W3fchV4D5umENyOiqFynLgiq0KJbnE7i+qS8axMO4KZeEUyoxwvWI2pYYycPdeLSF8OPKucnXSD4N++OtT9+Jbs44D9A4doR4K0Wd0gX6gSzRAzDvzxt7UE/5XP/Znfrpy9VpNzFu0Jf78L0ZOPVY=</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. 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=">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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>
  16. 16. 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
  17. 17. 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 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sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit 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sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit 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  18. 18. 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 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sha1_base64="oka71jUS/ZsLYG94jO48cIznUME=">AAAB83icbVDLSsNAFL2pr1pfVZduBovgqiQi6LLoxmUF+4CmlMl00g6dTMLMjVhCf8ONC0Xc+jPu/BsnbRbaemDgcM693DMnSKQw6LrfTmltfWNzq7xd2dnd2z+oHh61TZxqxlsslrHuBtRwKRRvoUDJu4nmNAok7wST29zvPHJtRKwecJrwfkRHSoSCUbSSr4kfURwHYfY0G1Rrbt2dg6wSryA1KNAcVL/8YczSiCtkkhrT89wE+xnVKJjks4qfGp5QNqEj3rNU0YibfjbPPCNnVhmSMNb2KSRz9fdGRiNjplFgJ/OEZtnLxf+8XorhdT8TKkmRK7Y4FKaSYEzyAshQaM5QTi2hTAublbAx1ZShraliS/CWv7xK2hd1z61795e1xk1RRxlO4BTOwYMraMAdNKEFDBJ4hld4c1LnxXl3PhajJafYOYY/cD5/ACqzkcI=</latexit> r ⌦ x = exp0(r log0(x))<latexit 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  19. 19. 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 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  20. 20. 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 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sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit> exp0(f(log0(x)))<latexit 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  21. 21. 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 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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 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sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit> f(x)<latexit 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sha1_base64="/8gM5rEdBwHBaq28EFv+YPEfZsk=">AAAB9HicbVDLSgMxFL2pr1pfVZdugkWomzIjgi6LblxWsA9oh5JJM21oJjMmmWIZ+h1uXCji1o9x59+YaWehrQcCh3Pu5Z4cPxZcG8f5RoW19Y3NreJ2aWd3b/+gfHjU0lGiKGvSSESq4xPNBJesabgRrBMrRkJfsLY/vs389oQpzSP5YKYx80IylDzglBgreUG1FxIz8oP0aXbeL1ecmjMHXiVuTiqQo9Evf/UGEU1CJg0VROuu68TGS4kynAo2K/USzWJCx2TIupZKEjLtpfPQM3xmlQEOImWfNHiu/t5ISaj1NPTtZBZRL3uZ+J/XTUxw7aVcxolhki4OBYnAJsJZA3jAFaNGTC0hVHGbFdMRUYQa21PJluAuf3mVtC5qrlNz7y8r9Zu8jiKcwClUwYUrqMMdNKAJFB7hGV7hDU3QC3pHH4vRAsp3juEP0OcPjFmR8Q==</latexit><latexit 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  22. 22. Hyperbolic RNN/GRU • Hyperbolic RNN • Hyperbolic GRU

  23. 23. Experiments
  24. 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. 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. 26. Results
  27. 27. Decision Boundary (SNLI) • No advantage for hyperbolic MLR in SNLI task
  28. 28. 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
  29. 29. More Details and Related Works https://tech-blog.abeja.asia/entry/hyperbolic_ml_2019

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