Robust Rank Estimation for Noisy Matrices

TL;DR

This paper introduces DICMR, a robust and computationally efficient criterion for estimating the true rank of noisy matrices, outperforming existing methods in accuracy and speed.

Contribution

The paper proposes DICMR, a new robust rank estimation criterion based on density power divergence, with proven asymptotic properties and practical advantages over existing methods.

Findings

DICMR achieves comparable accuracy to robust cross-validation methods.

DICMR has significantly lower computational cost.

DICMR outperforms state-of-the-art algorithms in microarray data imputation.

Abstract

Estimating the true rank of a noisy data matrix is a fundamental problem underlying techniques such as principal component analysis, matrix completion, etc. Existing rank estimation criteria, including information-based and cross-validation methods, are either highly sensitive to outliers or computationally demanding when combined with robust estimators. This paper proposes a new criterion, the Divergence Information Criterion for Matrix Rank (DICMR), that achieves both robustness and computational simplicity. Derived from the density power divergence framework, DICMR inherits the robustness properties while being computationally very simple. We provide asymptotic bounds on its overestimation and underestimation probabilities, and demonstrate first-order B-robustness of the criteria. Extensive simulations show that DICMR delivers accuracy comparable to the robustified cross-validation methods, but with far lower computational cost. We also showcase a real-data application to microarray imputation to further demonstrate its practical utility, outperforming several state-of-the-art algorithms.