Paper · Dec 21, 2021
Dynamically Stable Poincaré Embeddings for Neural Manifolds
As on arXiv
Abstract
In a Riemannian manifold, the Ricci flow is a partial differential equation for evolving the metric to become more regular. We hope that topological structures from such metrics may be used to assist in the tasks of machine learning. However, this part of the work is still missing. In this paper, we propose Ricci flow assisted Eucl2Hyp2Eucl neural networks that bridge this gap between the Ricci flow and deep neural networks by mapping neural manifolds from the Euclidean space to the dynamically stable Poincaré ball and then back to the Euclidean space. As a result, we prove that, if initial metrics have an -norm perturbation which deviates from the Hyperbolic metric on the Poincaré ball, the scaled Ricci-DeTurck flow of such metrics smoothly and exponentially converges to the Hyperbolic metric. Specifically, the role of the Ricci flow is to serve as naturally evolving to the stable Poincaré ball. For such dynamically stable neural manifolds under the Ricci flow, the convergence of neural networks embedded with such manifolds is not susceptible to perturbations. And we show that Ricci flow assisted Eucl2Hyp2Eucl neural networks outperform with their all Euclidean counterparts on image classification tasks.
- Submitted
- Dec 21, 2021
- Last revised
- Feb 16, 2022 (v2)
- Subjects
- cs.LG, math-ph
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 6D.11Computing Geometric Flows
Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.