Functorial Manifold Learning

Abstract

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 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 invariants. We express several popular manifold learning algorithms as functors at different levels of this hierarchy and present bounds on how closely the embeddings these algorithms produce from noisy data approximate the embeddings they would learn from noiseless data.

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Reproductions