Title: Flows on convex polytopes
Primary area: learning on graphs and other geometries & topologies
Keywords: convex polytopes, normalizing flows, Riemannian manifolds, continuous normalizing flows, flow matching, circular/cylinder spline flows, barycentric coordinates, maximum-entropy coordinates, Aitchison geometry, isometric log-ratio transform, constraint-based metabolic modeling, 13 C metabolic flux analysis, simulation-based inference

Abstract: 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 unique barycentric coordinates. Our experiments take inspiration from applications in constraint based metabolic modeling and demonstrate that our methods approximate exact sampling distributions and achieve fast training and inference times.

TL;DR: Modeling high-dimensional distributions on convex polytopes with normalizing flows via ball–polytope homeomorphisms and barycentric coordinates.