Optimal Rates of Statistical Seriation

TL;DR

This paper investigates the statistical limits of permuting noisy matrices to achieve monotonic or unimodal columns, proposing optimal estimators and analyzing their theoretical and practical performance.

Contribution

It introduces a minimax rate analysis for the seriation problem with noisy data and proposes efficient estimators that adapt to natural matrix structures.

Findings

Least squares estimator is near-optimal for monotone and unimodal columns.

Proposed efficient estimator performs well both theoretically and experimentally.

The work bridges shape-constrained estimation and permutation learning.

Abstract

Given a matrix the seriation problem consists in permuting its rows in such way that all its columns have the same shape, for example, they are monotone increasing. We propose a statistical approach to this problem where the matrix of interest is observed with noise and study the corresponding minimax rate of estimation of the matrices. Specifically, when the columns are either unimodal or monotone, we show that the least squares estimator is optimal up to logarithmic factors and adapts to matrices with a certain natural structure. Finally, we propose a computationally efficient estimator in the monotonic case and study its performance both theoretically and experimentally. Our work is at the intersection of shape constrained estimation and recent work that involves permutation learning, such as graph denoising and ranking.