On the maximal size of Large-Average and ANOVA-fit Submatrices in a Gaussian Random Matrix

TL;DR

This paper analyzes the maximum size of specific submatrices within Gaussian random matrices, providing thresholds and probability bounds for large-average and ANOVA-fit submatrices, supported by theoretical and simulation results.

Contribution

It introduces size thresholds and asymptotic probability bounds for large-average and ANOVA-fit submatrices in Gaussian matrices, including concentration results and simulations.

Findings

Identified size thresholds for submatrices with high average or ANOVA fit.

Derived asymptotic probability bounds for these submatrices.

Validated theoretical predictions with simulation results.

Abstract

We investigate the maximal size of distinguished submatrices of a Gaussian random matrix. Of interest are submatrices whose entries have average greater than or equal to a positive constant, and submatrices whose entries are well-fit by a two-way ANOVA model. We identify size thresholds and associated (asymptotic) probability bounds for both large-average and ANOVA-fit submatrices. Results are obtained when the matrix and submatrices of interest are square, and in rectangular cases when the matrix submatrices of interest have fixed aspect ratios. In addition, we obtain a strong, interval concentration result for the size of large average submatrices in the square case. A simulation study shows good agreement between the observed and predicted sizes of large average submatrices in matrices of moderate size.