Distributional limits of graph cuts on discretized grids

TL;DR

This paper derives the asymptotic distributional limits of various graph cut methods on discretized grids, revealing different behaviors for Cheeger Cut depending on the partition volumes, and validates these results through experiments and applications to the Xist algorithm.

Contribution

It provides explicit distributional limits for balanced graph cuts and extends the analysis to the Xist algorithm, advancing understanding of statistical inference in graph-based clustering.

Findings

Minimum Cut, Ratio Cut, and Normalized Cut behave as the minimum of Gaussians asymptotically.

Cheeger Cut's distribution depends on the volume of the optimal partition, showing a dichotomy.

Bootstrap consistency is established for all graph cut types.

Abstract

Graph cuts are among the most prominent tools for clustering and classification analysis. While intensively studied from geometric and algorithmic perspectives, graph cut-based statistical inference still remains elusive to a certain extent. Distributional limits are fundamental in understanding and designing such statistical procedures on randomly sampled data. We provide explicit limiting distributions for balanced graph cuts in general on a fixed but arbitrary discretization. In particular, we show that Minimum Cut, Ratio Cut and Normalized Cut behave asymptotically as the minimum of Gaussians as sample size increases. Interestingly, our results reveal a dichotomy for Cheeger Cut: The limiting distribution of the optimal objective value is the minimum of Gaussians only when the optimal partition yields two sets of unequal volumes, while otherwise the limiting distribution is the minimum of a random mixture of Gaussians. Further, we show the bootstrap consistency for all types of graph cuts by utilizing the directional differentiability of cut functionals. We validate these theoretical findings by Monte Carlo experiments, and examine differences between the cuts and the dependency on the underlying distribution. Additionally, we expand our theoretical findings to the Xist algorithm, a computational surrogate of graph cuts recently proposed in Suchan, Li and Munk (arXiv, 2023), thus demonstrating the practical applicability of our findings e.g. in statistical tests.