Spectral Algorithms Optimally Recover Planted Sub-structures

TL;DR

This paper demonstrates that spectral algorithms can universally achieve optimal solutions across various planted substructure detection problems, extending previous results beyond community detection in stochastic block models.

Contribution

It generalizes the optimality of spectral algorithms to multiple planted substructure problems, including dense subgraphs and submatrix localization, beyond prior specific cases.

Findings

Spectral algorithms are optimal for planted dense subgraph detection.

Optimality extends to submatrix localization problems.

Results apply to censored variants of these problems.

Abstract

Spectral algorithms are an important building block in machine learning and graph algorithms. We are interested in studying when such algorithms can be applied directly to provide optimal solutions to inference tasks. Previous works by Abbe, Fan, Wang and Zhong (2020) and by Dhara, Gaudio, Mossel and Sandon (2022) showed the optimality for community detection in the Stochastic Block Model (SBM), as well as in a censored variant of the SBM. Here we show that this optimality is somewhat universal as it carries over to other planted substructures such as the planted dense subgraph problem and submatrix localization problem, as well as to a censored version of the planted dense subgraph problem.