Functorial Manifold Learning

TL;DR

This paper introduces a functorial framework for manifold learning, characterizing algorithms as functors, analyzing their stability, and deriving new algorithms with competitive performance.

Contribution

It develops a categorical perspective on manifold learning, providing refinement bounds, stability analysis, and novel algorithms within a unified functorial framework.

Findings

Characterized manifold learning algorithms as functors from pseudometric spaces to optimization objectives.

Proved bounds on the loss functions of manifold learning algorithms.

Derived new manifold learning algorithms that perform competitively with existing methods.

Abstract

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and that factor through hierarchical clustering functors. We then use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their equivariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, and UMAP. Next, we use interleaving distance to study the stability of a broad class of manifold learning algorithms. We present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data. Finally, we use our framework to derive a set of novel manifold learning algorithms, which we experimentally demonstrate are competitive with the state of the art.