Comparing the Switch and Curveball Markov Chains for Sampling Binary Matrices with Fixed Marginals

TL;DR

This paper compares the spectral gaps of switch-based Markov chains and the Curveball chain for sampling binary matrices with fixed marginals, showing the Curveball chain's rapid mixing under certain conditions and extending analysis to matrices with forbidden entries.

Contribution

It provides a spectral gap comparison demonstrating the rapid mixing of the Curveball chain relative to switch chains and extends the analysis to matrices with forbidden entries.

Findings

Curveball chain is rapidly mixing if switch chains are rapidly mixing.

The switch Markov chain has non-negative eigenvalues for matrices with at least three columns.

Improved bounds on the smallest eigenvalue for sampling directed regular graphs.

Abstract

The Curveball algorithm is a variation on well-known switch-based Markov chain approaches for uniformly sampling binary matrices with fixed row and column sums. Instead of a switch, the Curveball algorithm performs a so-called binomial trade in every iteration of the algorithm. Intuitively, this could lead to a better convergence rate for reaching the stationary (uniform) distribution in certain cases. Some experimental evidence for this has been given in the literature. In this note we give a spectral gap comparison between two switch-based chains and the Curveball chain. In particular, this comparison allows us to conclude that the Curveball Markov chain is rapidly mixing whenever one of the two switch chains is rapidly mixing. Our analysis directly extends to the case of sampling binary matrices with forbidden entries (under the assumption of irreducibility). This in particular captures the case of sampling simple directed graphs with given degrees. As a by-product of our analysis, we show that the switch Markov chain of the Kannan-Tetali-Vempala conjecture only has non-negative eigenvalues if the sampled binary matrices have at least three columns. This shows that the Markov chain does not have to be made lazy, which is of independent interest. We also obtain an improved bound on the smallest eigenvalue for the switch Markov chain studied by Greenhill for uniformly sampling simple directed regular graphs.