A Discussion On the Validity of Manifold Learning
This work addresses a fundamental validity issue in manifold learning for researchers in machine learning and applied mathematics, though it is incremental as it highlights problems without fully solving them.
The paper identifies that widely used dimensionality reduction and manifold learning methods violate mathematical manifold definitions, failing to produce valid manifold representations. It proposes a provably correct algorithm, FPLM, with geometric guarantees for valid manifold learning up to a homeomorphism, but notes that constructing a fully bijective mapping remains an open problem.
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.