Functorial Manifold Learning

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.

Data science and data analytics in life science research

“…This might allow for further improvements of the methods. A possible avenue from a category-theoretical point of view might be to combine the theory developed so far with ideas from functorial manifold learning and clustering as investigated in [Shiebler, 2020b] and [Shiebler, 2020a]. Furthermore, it might be helpful to include the stochastic gradient descent procedure as well into the category-theoretical formulation, as initiated in [Fong et al, 2019] and [Spivak, 2022] (which in turn was building up on [Spivak and Niu, 2023]).…”

“…We also note that a general category-theoretical perspective on different manifold learning schemes and how they relate to clustering algorithms is given in the references [Shiebler, 2020b] and [Shiebler, 2020a]. Of those methods, UMAP is interesting from a category-theoretical point of view because the authors pick up and modify the adjunction Sing : UM sFuz : Re introduced in [Spivak, 2009] to justify (a substantial part of) their procedure.…”