Detection-Recovery and Detection-Refutation Gaps via Reductions from Planted Clique

TL;DR

This paper establishes detection-recovery and detection-refutation gaps in the Planted Dense Subgraph problem by leveraging reductions from a variant of the Planted Clique Hypothesis with secret leakage, advancing understanding of computational-statistical gaps.

Contribution

It introduces a new reduction framework from a modified Planted Clique Hypothesis to demonstrate detection-recovery and refutation gaps in PDS, providing sharp lower bounds.

Findings

Detection-recovery gap established under the secret leakage Planted Clique Hypothesis.

Detection-refutation gap demonstrated with sharp lower bounds.

Framework extends previous reduction techniques to average-case problems.

Abstract

Planted Dense Subgraph (PDS) problem is a prototypical problem with a computational-statistical gap. It also exhibits an intriguing additional phenomenon: different tasks, such as detection or recovery, appear to have different computational limits. A detection-recovery gap for PDS was substantiated in the form of a precise conjecture given by Chen and Xu (2014) (based on the parameter values for which a convexified MLE succeeds) and then shown to hold for low-degree polynomial algorithms by Schramm and Wein (2022) and for MCMC algorithms for Ben Arous et al. (2020). In this paper, we demonstrate that a slight variation of the Planted Clique Hypothesis with secret leakage (introduced in Brennan and Bresler (2020)), implies a detection-recovery gap for PDS. In the same vein, we also obtain a sharp lower bound for refutation, yielding a detection-refutation gap. Our methods build on the framework of Brennan and Bresler (2020) to construct average-case reductions mapping secret leakage Planted Clique to appropriate target problems.