Statistical and computational thresholds for the planted $k$-densest sub-hypergraph problem

TL;DR

This paper investigates the thresholds for exactly recovering a planted dense sub-hypergraph in a uniform hypergraph, revealing a statistical-to-computational gap and how signal structure influences phase transitions.

Contribution

It provides tight information-theoretic and algorithmic bounds for the recovery problem, highlighting the impact of signal structure on phase transitions, unlike previous tensor-PCA bounds.

Findings

Identifies exact recovery thresholds via maximum-likelihood estimation.

Demonstrates a widening statistical-to-computational gap with increased sparsity.

Shows the influence of signal structure on phase transition locations.

Abstract

In this work, we consider the problem of recovery a planted $k$-densest sub-hypergraph on $d$-uniform hypergraphs. This fundamental problem appears in different contexts, e.g., community detection, average-case complexity, and neuroscience applications as a structural variant of tensor-PCA problem. We provide tight \emph{information-theoretic} upper and lower bounds for the exact recovery threshold by the maximum-likelihood estimator, as well as \emph{algorithmic} bounds based on approximate message passing algorithms. The problem exhibits a typical statistical-to-computational gap observed in analogous sparse settings that widen with increasing sparsity of the problem. The bounds show that the signal structure impacts the location of the statistical and computational phase transition that the known existing bounds for the tensor-PCA model do not capture. This effect is due to the generic planted signal prior that this latter model addresses.