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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
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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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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
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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>
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=">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})
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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 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
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

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Hyperbolic Neural Networks

  • 1. Hyperbolic Neural Networks 2019 Jan. 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 my blog 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 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 <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 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>
  • 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=">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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=">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><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 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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. 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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  • 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. 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. 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 sha1_base64="7SVi6ojst7Ysbg69G/olY6DNddU=">AAACHnicbVDLSgMxFM34rPU16tJNsAjtpsyIohuh6MZlBfuAzjBk0kwbmkmGJCMtQ7/Ejb/ixoUigiv9GzNtQW09EDiccy8594QJo0o7zpe1tLyyurZe2Chubm3v7Np7+00lUolJAwsmZDtEijDKSUNTzUg7kQTFISOtcHCd+617IhUV/E6PEuLHqMdpRDHSRgrsMwk9oWlMFPRipPthlA3H8BJ6ZJgEThkam4le4MDyj12pBHbJqToTwEXizkgJzFAP7A+vK3AaE64xQ0p1XCfRfoakppiRcdFLFUkQHqAe6RjKkQnkZ5PzxvDYKF0YCWke13Ci/t7IUKzUKA7NZJ5RzXu5+J/XSXV04WeUJ6kmHE8/ilIGtYB5V7BLJcGajQxBWFKTFeI+kghr02jRlODOn7xImidV16m6t6el2tWsjgI4BEegDFxwDmrgBtRBA2DwAJ7AC3i1Hq1n6816n44uWbOdA/AH1uc3FKShLg==</latexit><latexit 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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 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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. Hyperbolic RNN/GRU • Hyperbolic RNN • Hyperbolic GRU

  • 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
  • 27. Decision Boundary (SNLI) • No advantage for hyperbolic MLR in SNLI task
  • 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