PolytopeWalk: Sparse MCMC Sampling over Polytopes
Benny Sun, Yuansi Chen
TL;DR
PolytopeWalk tackles high-dimensional uniform sampling over polytopes by marrying barrier-based interior-point MCMC with sparse, constrained formulations. The authors implement four barrier-based walks (Dikin, Vaidya, John, Lee-Sidford) along with Ball Walk and Hit-and-Run, and provide both dense full-dimensional and sparse constrained variants, all within an open-source C++/Python package. A core contribution is the facial reduction framework, which removes degeneracy and yields reduced-dimension representations that preserve sparsity and numerical stability, enabling scalable sampling up to dimensions exceeding $10^5$. Comprehensive preprocessing (initialization and facial reduction) and rigorous comparisons against Volesti demonstrate improved per-iteration cost and mixing on Netlib datasets and structured polytopes, validating the approach for applications in Bayesian uncertainty quantification, systems biology, and volume computation. The work thus delivers a scalable, well-documented toolkit for high-dimensional, sparse polytope sampling with strong practical impact.
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 .