A Discussion On the Validity of Manifold Learning

TL;DR

This paper critically examines the validity of common manifold learning methods, revealing fundamental issues with their mathematical assumptions and proposing a new algorithm with geometric guarantees for valid manifold representation.

Contribution

It identifies key limitations in existing manifold learning techniques and introduces the fixed points Laplacian mapping (FPLM), a provably correct algorithm with geometric guarantees.

Findings

Existing methods violate manifold mapping assumptions

FPLM provides a valid manifold representation with geometric guarantees

Constructing bijective mappings remains an open mathematical problem

Abstract

Dimensionality reduction (DR) and manifold learning (ManL) have been applied extensively in many machine learning tasks, including signal processing, speech recognition, and neuroinformatics. However, the understanding of whether DR and ManL models can generate valid learning results remains unclear. In this work, we investigate the validity of learning results of some widely used DR and ManL methods through the chart mapping function of a manifold. We identify a fundamental problem of these methods: the mapping functions induced by these methods violate the basic settings of manifolds, and hence they are not learning manifold in the mathematical sense. To address this problem, we provide a provably correct algorithm called fixed points Laplacian mapping (FPLM), that has the geometric guarantee to find a valid manifold representation (up to a homeomorphism). Combining one additional condition(orientation preserving), we discuss a sufficient condition for an algorithm to be bijective for any d-simplex decomposition result on a d-manifold. However, constructing such a mapping function and its computational method satisfying these conditions is still an open problem in mathematics.