Seriation of Toeplitz and latent position matrices: optimal rates and computational trade-offs

TL;DR

This paper investigates the problem of ordering structured matrices, specifically Toeplitz and latent position matrices, under noisy observations, revealing a statistical-computational gap and providing optimal algorithms and bounds.

Contribution

It introduces a polynomial-time algorithm achieving the optimal statistical rate and characterizes the information-theoretic limits for seriation of structured matrices.

Findings

Identifies a statistical-computational gap in seriation tasks.

Provides a polynomial-time algorithm matching the lower bound.

Characterizes the optimal risk for matrix seriation.

Abstract

In this paper, we consider the problem of seriation of a permuted structured matrix based on noisy observations. The entries of the matrix relate to an expected quantification of interaction between two objects: the higher the value, the closer the objects. A popular structured class for modelling such matrices is the permuted Robinson class, namely the set of matrices whose coefficients are decreasing away from its diagonal, up to a permutation of its lines and columns. We consider in this paper two submodels of Robinson matrices: the Toeplitz model, and the latent position model. We provide a computational lower bound based on the low-degree paradigm, which hints that there is a statistical-computational gap for seriation when measuring the error based on the Frobenius norm. We also provide a simple and polynomial-time algorithm that achives this lower bound. Along the way, we also characterize the information-theory optimal risk thereby giving evidence for the extent of the computation/information gap for this problem.