Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dynamically Stable Poincar\'e Embeddings for Neural Manifolds   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 Poincar\'e ball and then
back to the Euclidean space. As a result, we prove that, if initial metrics
have an $L^2$-norm perturbation which deviates from the Hyperbolic metric on
the Poincar\'e 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 Poincar\'e 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.

9
The categories of this paper are: nlin
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th