Efficient computation of the volume of a polytope in high-dimensions using Piecewise Deterministic Markov Processes

TL;DR

This paper introduces a novel sampling method based on Piecewise Deterministic Markov Processes for efficiently computing high-dimensional polytope volumes, significantly reducing computational costs compared to existing Hamiltonian Monte Carlo methods.

Contribution

The paper proposes a new PDM process-based sampling strategy that simplifies simulation and accelerates volume computation in high-dimensional polytopes.

Findings

Method is one order of magnitude faster than Hamiltonian Monte Carlo.

Algorithm is numerically robust up to dimension 500.

Reduces computational cost by a factor of the dimension.

Abstract

Computing the volume of a polytope in high dimensions is computationally challenging but has wide applications. Current state-of-the-art algorithms to compute such volumes rely on efficient sampling of a Gaussian distribution restricted to the polytope, using e.g. Hamiltonian Monte Carlo. We present a new sampling strategy that uses a Piecewise Deterministic Markov Process. Like Hamiltonian Monte Carlo, this new method involves simulating trajectories of a non-reversible process and inherits similar good mixing properties. However, importantly, the process can be simulated more easily due to its piecewise linear trajectories - and this leads to a reduction of the computational cost by a factor of the dimension of the space. Our experiments indicate that our method is numerically robust and is one order of magnitude faster (or better) than existing methods using Hamiltonian Monte Carlo. On a single core processor, we report computational time of a few minutes up to dimension 500.