Discrete Approximate Circle Bundles For Data

We introduce discrete approximate circle bundles, a class of objects designed to serve as the data science analog of circle bundles from algebraic topology. We show that, under appropriate conditions, one can meaningfully and stably identify a discrete approximate circle bundle with an isomorphism class of true circle bundles. We also describe two cohomology invariants which uniquely determine the isomorphism class of a circle bundle, and provide algorithms to compute them given a discrete approximate representative. Finally, we propose a novel methodology for coordinatization and dimensionality reduction of circle bundle data. We apply these tools to study a benchmark optical flow dataset, where we confirm the toroidal model proposed by Adams et al. and discover larger spaces in other density regimes. This is joint work with Jose A. Perea.