Flows on convex polytopes

TL;DR

This work develops a principled framework to model high-dimensional distributions supported on convex polytopes by leveraging a ball-homeomorphism that converts polytopes to the unit ball, enabling efficient flow-based modeling on a manifold. It extends discrete and continuous normalizing flows to Riemannian settings, introduces a hit-and-run–based ball transform, and adapts spline and flow-matching techniques to balls and polytopes. A key contribution is a V-representation–friendly construction using maximum entropy barycentric coordinates and Aitchison geometry, allowing density estimation and sampling without geometry conversion to the H-representation. Empirical results in metabolic flux analysis demonstrate competitive density estimation, accurate sampling, and fast training/inference, highlighting practical scalability and flexibility for flux-analysis–style problems. Overall, the approach provides an amortized, geometry-respecting framework for probabilistic modeling on polytopes with clear pathways for high-dimensional extensions and Bayesian design.

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 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.

Figures (4)

• Figure 1: Graphical intuition for the homeomorphism between a 2D convex polytope and the disk.
• Figure 2: Yellow: the John polytope (projected onto 2D). Red: convex hull of all samples from either the flow or target. Kernel density estimates (from 10k samples) are overlaid, with red points indicating the means of the three Gaussians in the target density $p^{mog}_\mathcal{F}$. (A) 105k samples from MCMC; (B) 20k samples from a cylinder spline flow $q^{spline}$; (C) 125k samples from the uniform target $p^{\mathcal{U}}_\mathcal{F}$ (via MCMC); (D) 20k samples from a Euclidean CNF $q^{eucl}$; (E) 20k samples from a Riemannian CNF $q^{ball}$; (F) 20k samples from a Euclidean CNF on standardized, projected $\mathop{\mathrm{ilr}}\nolimits$ coordinates $q^{ait}$.
• Figure 3: Similar to Figure \ref{['fig:flows_fig']}, but here $p^{mog}_\square$ is the target. (A) 125k MCMC samples; (C) 20k samples from the Euclidean CNF $q^{eucl}$.
• Figure : Multiple proposal Hit-and-Run sampling of distributions with polytope support