Asymptotic Bounds and Online Algorithms for Average-Case Matrix Discrepancy

TL;DR

This work advances matrix discrepancy by (i) characterizing the asymptotic discrepancy for GOE matrices in the average-case regime, showing tight concentration around $\frac{2}{e^{3/4}}\sqrt{nm}\,4^{-\xi n/m^2}$ under $m^2\ll n/\log n$ and establishing a matching upper bound for $n=\Omega(m^2)$, and (ii) analyzing the online matrix hyperbolic cosine (MHC) algorithm under broad random matrix models, proving it achieves discrepancy $O(m\log m)$ with high probability. The analysis combines first- and second-mmoment methods, Laplace-type asymptotics, Gramian spectral techniques, and subgaussian/vectorization tools, and extends results to general random matrices and Wishart ensembles. These results illuminate strong cancellations in random matrix sums and provide robust online-discrepancy guarantees under realistic randomness assumptions, connecting discrepancy theory with random matrix theory and online algorithms. The findings have implications for spectral balancing, graph sparsification, and uncertainty reduction in high-dimensional random settings.

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 $Θ(\sqrt{nm}4^{-(1 + o(1))n/m^2})$, 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 $n = Ω(m^2)$, 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.

Theorems & Definitions (62)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 1: Equation 2.5.1 in Anderson10
• Lemma 2: Theorem 2.5.2 in Anderson10
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6