Universal Rank Inference via Residual Subsampling with Application to Large Networks

TL;DR

This paper introduces a universal residual subsampling method for accurately inferring the rank of large matrices, including network models, with strong theoretical guarantees and demonstrated effectiveness through simulations and real data.

Contribution

It proposes a novel residual subsampling approach for rank inference that works across various models, including complex network models, with proven asymptotic properties.

Findings

The RIRS method accurately tests and estimates matrix rank.

The test statistic follows a normal distribution under the null hypothesis.

The method performs well in simulations and real data applications.

Abstract

Determining the precise rank is an important problem in many large-scale applications with matrix data exploiting low-rank plus noise models. In this paper, we suggest a universal approach to rank inference via residual subsampling (RIRS) for testing and estimating rank in a wide family of models, including many popularly used network models such as the degree corrected mixed membership model as a special case. Our procedure constructs a test statistic via subsampling entries of the residual matrix after extracting the spiked components. The test statistic converges in distribution to the standard normal under the null hypothesis, and diverges to infinity with asymptotic probability one under the alternative hypothesis. The effectiveness of RIRS procedure is justified theoretically, utilizing the asymptotic expansions of eigenvectors and eigenvalues for large random matrices recently developed in [11] and [12]. The advantages of the newly suggested procedure are demonstrated through several simulation and real data examples.