# A Fast MCMC for the Uniform Sampling of Binary Matrices with Fixed Margins

## TL;DR

This paper introduces the Rectangle Loop algorithm, a Markov chain Monte Carlo method that efficiently and uniformly samples binary matrices with fixed margins, outperforming traditional swap algorithms especially for large or sparse/dense matrices.

## Contribution

The paper presents the Rectangle Loop algorithm, a novel MCMC approach that improves efficiency and theoretical properties over existing swap algorithms for binary matrix sampling.

## Key findings

- Rectangle Loop algorithm is more efficient than swap algorithm.
- Theoretically superior in Peskun's order.
- Empirical results confirm improved performance for large or sparse/dense matrices.

## Abstract

Uniform sampling of binary matrix with fixed margins is an important and difficult problem in statistics, computer science, ecology and so on. The well-known swap algorithm would be inefficient when the size of the matrix becomes large or when the matrix is too sparse/dense. Here we propose the Rectangle Loop algorithm, a Markov chain Monte Carlo algorithm to sample binary matrices with fixed margins uniformly. Theoretically the Rectangle Loop algorithm is better than the swap algorithm in Peskun's order. Empirically studies also demonstrates the Rectangle Loop algorithm is remarkablely more efficient than the swap algorithm.