Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms

TL;DR

This paper studies the problem of optimally reordering noisy matrices, establishing fundamental limits, analyzing existing algorithms, and proposing a new efficient method that outperforms current approaches, with applications in biology.

Contribution

It introduces a novel polynomial-time adaptive sorting algorithm with guaranteed performance improvements for matrix reordering tasks.

Findings

The constrained least squares estimator achieves the optimal rate.

Spectral seriation is suboptimal for noisy disordered matrices.

The proposed adaptive sorting algorithm outperforms existing methods in simulations and real data.

Abstract

Motivated by applications in single-cell biology and metagenomics, we investigate the problem of matrix reordering based on a noisy disordered monotone Toeplitz matrix model. We establish the fundamental statistical limit for this problem in a decision-theoretic framework and demonstrate that a constrained least squares estimator achieves the optimal rate. However, due to its computational complexity, we analyze a popular polynomial-time algorithm, spectral seriation, and show that it is suboptimal. To address this, we propose a novel polynomial-time adaptive sorting algorithm with guaranteed performance improvement. Simulations and analyses of two real single-cell RNA sequencing datasets demonstrate the superiority of our algorithm over existing methods.