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Robust Rank Estimation for Noisy Matrices

Subhrajyoty Roy, Abhik Ghosh, Ayanendranath Basu

TL;DR

The paper addresses robustly estimating the true rank of a noisy data matrix by modeling it as a low-rank component plus sparse outliers and dense noise, and introduces a Divergence Information Criterion for Matrix Rank (DICMR) that leverages density power divergence with a robust SVD backbone. It establishes selection consistency and first-order robustness for DICMR, and demonstrates through extensive simulations that it rivals robust cross-validation at substantially lower computational cost. The method is validated on real data from microarray imputation (RNA-seq context), where DICMR-guided robust rank estimation yields improved reconstruction performance, highlighting practical significance for high-dimensional biological data analysis. While DICMR may overestimate rank for nearly square matrices, it provides a fast, reliable upper-bound estimator that can be integrated with more exhaustive search strategies or tensor generalizations in future work.

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.

Robust Rank Estimation for Noisy Matrices

TL;DR

The paper addresses robustly estimating the true rank of a noisy data matrix by modeling it as a low-rank component plus sparse outliers and dense noise, and introduces a Divergence Information Criterion for Matrix Rank (DICMR) that leverages density power divergence with a robust SVD backbone. It establishes selection consistency and first-order robustness for DICMR, and demonstrates through extensive simulations that it rivals robust cross-validation at substantially lower computational cost. The method is validated on real data from microarray imputation (RNA-seq context), where DICMR-guided robust rank estimation yields improved reconstruction performance, highlighting practical significance for high-dimensional biological data analysis. While DICMR may overestimate rank for nearly square matrices, it provides a fast, reliable upper-bound estimator that can be integrated with more exhaustive search strategies or tensor generalizations in future work.

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.
Paper Structure (22 sections, 3 theorems, 57 equations, 4 figures, 11 tables)

This paper contains 22 sections, 3 theorems, 57 equations, 4 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Mathematical Description of the Problem
  3. Organization of the paper
  4. Existing Literature
  5. Penalization Methods
  6. Cross-Validation Approaches
  7. Other Approaches
  8. Divergence Information Criterion for Matrix Rank
  9. Theoretical Studies
  10. Selection Consistency
  11. Robustness
  12. Simulation Studies
  13. Simulation Setups
  14. Comparative Algorithms
  15. Simulation Results
  16. ...and 7 more sections

Key Result

Theorem 1

Let us assume that the Assumptions (B1)-(B6) of roy2024robustsvd hold with the density $g$ replaced by the model density $f$. Additionally, assume that Then, for any $\alpha > 0$, we have

Figures (4)

  • Figure 1: Probability of selecting the correct model (rank zero) as a function of the log-ratio $\log(n/p)$ for different values of $\alpha$ in the case of normally distributed errors.
  • Figure 2: A partially observed PANCAN single-cell RNA sequencing dataset, with the gray shaded region denoting unavailable part of the data. Each column is preprocessed using log2 transformation and normalization.
  • Figure 3: The imputed RNA-seq PANCAN dataset obtained by different imputation techniques; the dashed line shows the partition for which the bottom-right corner block is imputed; "norm" in brackets indicate a preprocssing normalization step is performed.
  • Figure 4: Estimated rank and relative MSE based on DICMR and rSVDdpd algorithm

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3