Computational lower bounds in latent models: clustering, sparse-clustering, biclustering

TL;DR

This paper develops a new method to establish computational lower bounds in high-dimensional latent models, revealing gaps between statistical and computational performance in clustering, sparse clustering, and biclustering.

Contribution

It introduces a novel scheme for deriving lower bounds on low-degree polynomial performance, providing sharper results and simplified proofs for latent space models.

Findings

Established new computational lower bounds for clustering, sparse clustering, and biclustering.

Proved matching upper bounds and statistical results for the studied problems.

Revealed significant statistical-computational gaps in high-dimensional latent models.

Abstract

In many high-dimensional problems, like sparse-PCA, planted clique, or clustering, the best known algorithms with polynomial time complexity fail to reach the statistical performance provably achievable by algorithms free of computational constraints. This observation has given rise to the conjecture of the existence, for some problems, of gaps -- so called statistical-computational gaps -- between the best possible statistical performance achievable without computational constraints, and the best performance achievable with poly-time algorithms. A powerful approach to assess the best performance achievable in poly-time is to investigate the best performance achievable by polynomials with low-degree. We build on the seminal paper of Schramm and Wein (2022) and propose a new scheme to derive lower bounds on the performance of low-degree polynomials in some latent space models. By better leveraging the latent structures, we obtain new and sharper results, with simplified proofs. We then instantiate our scheme to provide computational lower bounds for the problems of clustering, sparse clustering, and biclustering. We also prove matching upper-bounds and some additional statistical results, in order to provide a comprehensive description of the statistical-computational gaps occurring in these three problems.