Computational barriers for permutation-based problems, and cumulants of weakly dependent random variables

TL;DR

This paper develops a method to bound cumulants in problems with weak dependencies, like permutations, revealing computational-statistical gaps in feature matching and seriation tasks.

Contribution

It introduces a novel technique for bounding cumulants under weak dependencies, extending analysis to permutation-based problems previously intractable.

Findings

Evidence of statistical-computational gaps in feature matching

Evidence of gaps in seriation problems

Method applicable to problems with dependent latent variables

Abstract

In many high-dimensional problems,polynomial-time algorithms fall short of achieving the statistical limits attainable without computational constraints. A powerful approach to probe the limits of polynomial-time algorithms is to study the performance of low-degree polynomials. The seminal work of arXiv:2008.02269 connects low-degree lower bounds to multivariate cumulants. Prior works arXiv:2308.15728, arXiv:2506.13647 leverage independence among latent variables to bound cumulants. However, such approaches break down for problems with latent structure lacking independence, such as those involving random permutations. To address this important restriction, we develop a technique to upper-bound cumulants under weak dependencies, such as those arising from sampling without replacement or random permutations. To show-case the effectiveness of our approach, we uncover evidence of statistical-computational gaps in multiple feature matching and in seriation problems.