More Related Content Similar to Hyperbolic Neural Networks (6) More from Tatsuya Shirakawa (13) Hyperbolic Neural Networks2. 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
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 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. 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
• 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. 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
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. 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. 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
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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
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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=">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</latexit><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>
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. 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. 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
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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
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})
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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 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
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
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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
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
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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>
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
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. More Details and Related Works
https://tech-blog.abeja.asia/entry/hyperbolic_ml_2019