Flows on convex polytopes
This provides a method for applications like metabolic flux analysis, but appears incremental as it adapts existing flow techniques to a specific geometric setting.
The paper tackles the problem of modeling complex distributions on convex polytopes by developing a framework that maps flows from a unit ball to polytopes, achieving competitive density estimation and sampling accuracy with fast training and inference times.
We present a framework for modeling complex, high-dimensional distributions on convex polytopes by leveraging recent advances in discrete and continuous normalizing flows on Riemannian manifolds. We show that any full-dimensional polytope is homeomorphic to a unit ball, and our approach harnesses flows defined on the ball, mapping them back to the original polytope. Furthermore, we introduce a strategy to construct flows when only the vertex representation of a polytope is available, employing maximum entropy barycentric coordinates and Aitchison geometry. Our experiments take inspiration from applications in metabolic flux analysis and demonstrate that our methods achieve competitive density estimation, sampling accuracy, as well as fast training and inference times.