Universality of Computational Lower Bounds for Submatrix Detection

Matthew Brennan, Guy Bresler, Wasim Huleihel

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


In the general submatrix detection problem, the task is to detect the presence of a small k×k submatrix with entries sampled from a distribution P in an n×n matrix of samples from Q. This formulation includes a number of well-studied problems, such as biclustering when P and Q are Gaussians and the planted dense subgraph formulation of community detection when the submatrix is a principal minor and P and Q are Bernoulli random variables. These problems all seem to exhibit a universal phenomenon: there is a statistical-computational gap depending on P and Q between the minimum k at which this task can be solved and the minimum k at which it can be solved in polynomial time. Our main result is to tightly characterize this computational barrier as a tradeoff between k and the KL divergences between P and Q through average-case reductions from the planted clique conjecture. These computational lower bounds hold given mild assumptions on P and Q arising naturally from classical binary hypothesis testing. Our results recover and generalize the planted clique lower bounds for Gaussian biclustering in Ma and Wu (2015) and Brennan et al. (2018) and for the sparse and general regimes of planted dense subgraph in Hajek et al. (2015) and Brennan et al. (2018). This yields the first universality principle for computational lower bounds obtained through average-case reductions.
Original languageEnglish
Title of host publicationProceedings of the Thirty-Second Conference on Learning Theory
EditorsAlina Beygelzimer, Daniel Hsu
Place of PublicationPhoenix, USA
Number of pages52
StatePublished - 1 Jan 2019
Event32nd Annual Conference on Learning Theory, COLT 2019 - Phoenix, United States
Duration: 25 Jun 201928 Jun 2019
Conference number: 32

Publication series

NameProceedings of Machine Learning Research
ISSN (Electronic)2640-3498


Conference32nd Annual Conference on Learning Theory, COLT 2019
Abbreviated titleCOLT 2019
Country/TerritoryUnited States


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