# A Discussion On the Validity of Manifold Learning

arXiv: https://arxiv.org/abs/2106.01608 · HTML: https://mlsanity.com/paper/2106.01608/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 38 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 2 of 2)

- [Delaunay triangulation of manifolds](https://mlsanity.com/paper/1311.0117/) (2013)
- [Bijective Mappings Of Meshes With Boundary And The Degree In Mesh Processing](https://mlsanity.com/paper/1310.0955/) (2013)

## Built on by (most cited first, 2 of 2)

- [Are We Using Autoencoders in a Wrong Way?](https://mlsanity.com/paper/2309.01532/) (2023)
- [Partially Linear Autoencoders for Manifold Learning and Dimensionality Reduction](https://mlsanity.com/paper/2608.29867/) (2026)

## Abstract (arXiv)

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.