Functorial Manifold Learning

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning. We first characterize manifold learning algorithms as functors that map pseudometric spaces to optimization objectives and factor through hierachical 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 invariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy, including Metric Multidimensional Scaling, IsoMap, Laplacian Eigenmaps, 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.