PolytopeWalk: Sparse MCMC Sampling over Polytopes
This provides an end-to-end solution for uniform sampling over polytopes, addressing applications in statistics, systems biology, and volume computation, but it is incremental as it builds on existing MCMC algorithms with sparse adaptations.
The authors tackled the problem of high-dimensional uniform sampling over polytopes by developing PolytopeWalk, a scalable Python library that implements six state-of-the-art MCMC algorithms and introduces novel sparse constrained formulations, demonstrating improved sampling efficiency and per-iteration cost on datasets like Netlib and structured polytopes, with scalability to dimensions over 10^5.
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 .