PolytopeWalk: Sparse MCMC Sampling over Polytopes

TL;DR

PolytopeWalk is a scalable Python library that enables efficient uniform sampling over high-dimensional polytopes using advanced MCMC algorithms and sparsity-preserving formulations, facilitating applications in statistics and systems biology.

Contribution

The paper introduces PolytopeWalk, a comprehensive library with novel sparse formulations of MCMC algorithms for high-dimensional polytope sampling, improving efficiency and scalability.

Findings

Enhanced sampling efficiency demonstrated on Netlib datasets.

Maintains sparsity for scalability to dimensions over 10^5.

Implementations outperform existing methods in high-dimensional settings.

Abstract

High dimensional sampling is an important computational tool in statistics and other computational disciplines, with applications ranging from Bayesian statistical uncertainty quantification, metabolic modeling in systems biology to volume computation. We present $\textsf{PolytopeWalk}$, a new scalable Python library designed for uniform sampling over polytopes. The library provides an end-to-end solution, which includes preprocessing algorithms such as facial reduction and initialization methods. Six state-of-the-art MCMC algorithms on polytopes are implemented, including the Dikin, Vaidya, and John Walk. Additionally, we introduce novel sparse constrained formulations of these algorithms, enabling efficient sampling from sparse polytopes of the form $K_2 = \{x \in \mathbb{R}^d \ | \ Ax = b, x \succeq_k 0\}$. This implementation maintains sparsity in $A$, ensuring scalability to high dimensional settings $(d > 10^5)$. We demonstrate the improved sampling efficiency and per-iteration cost on both Netlib datasets and structured polytopes. $\textsf{PolytopeWalk}$ is available at github.com/ethz-randomwalk/polytopewalk with documentation at polytopewalk.readthedocs.io .