Asymptotic Bounds and Online Algorithms for Average-Case Matrix Discrepancy

TL;DR

This paper characterizes the asymptotic behavior of matrix discrepancy for random matrices and analyzes the performance of an online discrepancy minimization algorithm in average-case scenarios.

Contribution

It provides a sharp asymptotic characterization of matrix discrepancy for Gaussian orthogonal ensemble matrices and analyzes the average-case performance of the matrix hyperbolic cosine algorithm.

Findings

Asymptotic discrepancy concentrates around a specific bound for Gaussian matrices.

The trivial bound is tight when the number of matrices is much less than the square of their size.

The online algorithm achieves discrepancy of O(m log m) for broad classes of random matrices.

Abstract

We study the matrix discrepancy problem in the average-case setting. Given a sequence of $m \times m$ symmetric matrices $A_1,\ldots,A_n$, its discrepancy is defined as the minimal spectral norm over all signed sums $\sum_{i=1}^n x_iA_i$ with $x_1,\ldots,x_n \in \{\pm1\}$. Our contributions are twofold. First, we study the asymptotic discrepancy of random matrices. When the matrices belong to the Gaussian orthogonal ensemble, we provide a sharp characterization of the asymptotic discrepancy and show that the limiting distribution is concentrated around $\Theta(\sqrt{nm}4^{-(1 + o(1))n/m^2})$, under the assumption $m^2 \ll n/\log{n}$. We observe that the trivial bound $O(\sqrt{nm})$ cannot be improved when $n \ll m^2$ and show that this phenomenon occurs for a broad class of random matrices. In the case $n = \Omega(m^2)$, we provide a matching upper bound. Second, we analyse the matrix hyperbolic cosine algorithm, an online algorithm for matrix discrepancy minimization due to Zouzias (2011), in the average-case setting. We show that the algorithm achieves with high probability a discrepancy of $O(m\log{m})$ for a broad class of random matrices, including Wigner matrices with entries satisfying a hypercontractive inequality and Gaussian Wishart matrices.