slides describing the following paper:
Octavian-Eugen Ganea et. al., "Hyperbolic Neural Networks", NeurIPS 2018
(NeurIPS 2018 paper reading time at PFN at 2019 Jan. 26th)
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
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=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit><latexit sha1_base64="q1GI/HvmbGGAJjf0U6VDrYNKi0M=">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</latexit>
15. More Gyrovector space
• Gyrovector space can be also defined for other hyperbolic models
- Einstein gyrovector space <—> algebraic formulation of special relativity
• Relation to Riemannian hyperbolic geometry
-
• Mid point
• Gyrotrigonometry
…
a ( a b) ⌦ t ! geodesic from a to b<latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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</latexit><latexit sha1_base64="6qRC2ZO3hLgZPEmz+DGFhgHnX1U=">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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
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?
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