Curvature-aware Manifold Learning

TL;DR

This paper introduces CAML, a curvature-aware manifold learning algorithm that incorporates curvature information to better handle non-isometric manifolds, improving stability over traditional methods.

Contribution

CAML is a novel manifold learning method that relaxes the local isometry assumption by integrating curvature information, enabling better analysis of general manifolds.

Findings

CAML outperforms traditional algorithms in neighborhood preservation.

CAML demonstrates increased stability in manifold learning tasks.

Incorporating curvature improves the handling of non-isometric manifolds.

Abstract

Traditional manifold learning algorithms assumed that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, they divided manifold into a set of overlapping local patches which are locally isometric to linear subsets of Euclidean space. By analyzing the global or local isometry assumptions it can be shown that the learnt manifold is a flat manifold with zero Riemannian curvature tensor. In general, manifolds may not satisfy these hypotheses. One major limitation of traditional manifold learning is that it does not consider the curvature information of manifold. In order to remove these limitations, we present our curvature-aware manifold learning algorithm called CAML. The purpose of our algorithm is to break the local isometry assumption and to reduce the dimension of the general manifold which is not isometric to Euclidean space. Thus, our method adds the curvature information to the process of manifold learning. The experiments have shown that our method CAML is more stable than other manifold learning algorithms by comparing the neighborhood preserving ratios.