A weighted message-passing algorithm to estimate volume-related properties of random polytopes

TL;DR

This paper introduces a weighted message-passing algorithm for estimating volume-related properties of random polytopes, offering an alternative to traditional approximation methods and demonstrating its effectiveness through numerical comparisons.

Contribution

It presents a novel weighted message-passing algorithm that directly handles entire functions for volume estimation of random polytopes, improving upon existing approximation techniques.

Findings

The algorithm accurately estimates volume-related properties.

Numerical results outperform traditional Hit-and-Run methods.

The approach is versatile for different implementations.

Abstract

In this letter, we introduce a novel message-passing algorithm for a class of problems which can be mathematically understood as estimating volume-related properties of random polytopes. Unlike the usual approach consisting in approximating the real-valued cavity marginal distributions by a few parameters, we propose a weighted message-passing algorithm to deal with the entire function. Various alternatives of how to implement our approach are discussed and numerical results for random polytopes are compared with results using the Hit-and-Run algorithm.