Manifold Micro-Surgery with Linearly Nearly Euclidean Metrics

Jun Chen, Tianxin Huang, Wenzhou Chen, Yong Liu

Keywords: Geometry, Ricci flow, Neural network, Metric learning, Information geometry

Abstract: The Ricci flow is a method of manifold surgery, which can trim manifolds to more regular. However, in most cases, the Rich flow tends to develop singularities and lead to divergence of the solution. In this paper, we propose linearly nearly Euclidean metrics to assist manifold micro-surgery, which means that we prove the dynamical stability and convergence of such metrics under the Ricci-DeTurck flow. From the information geometry and mirror descent points of view, we give the approximation of the steepest descent gradient flow on the linearly nearly Euclidean manifold with dynamical stability. In practice, the regular shrinking or expanding of Ricci solitons with linearly nearly Euclidean metrics will provide a geometric optimization method for the solution on a manifold.

One-sentence Summary: Optimize a linearly nearly Euclidean manifold with the Ricci flow