A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent

TL;DR

This paper introduces a novel framework using mirror descent to address matrix discrepancy problems, providing improved bounds for matrix Spencer conjecture variants and generalizing to Schatten norms.

Contribution

It develops a new approach for matrix discrepancy bounds via mirror descent, solving special cases of the matrix Spencer conjecture and extending results to Schatten norms.

Findings

Efficiently finds colorings with low operator norm discrepancy for low-rank matrices.

Reduces matrix Spencer conjecture to quantum entropy net existence on the spectraplex.

Generalizes discrepancy bounds to Schatten norms with explicit bounds.

Abstract

Motivated by the Matrix Spencer conjecture, we study the problem of finding signed sums of matrices with a small matrix norm. A well-known strategy to obtain these signs is to prove, given matrices $A_1, \dots, A_n \in \mathbb{R}^{m \times m}$, a Gaussian measure lower bound of $2^{-O(n)}$ for a scaling of the discrepancy body $\{x \in \mathbb{R}^n: \| \sum_{i=1}^n x_i A_i\| \leq 1\}$. We show this is equivalent to covering its polar with $2^{O(n)}$ translates of the cube $\frac{1}{n} B^n_\infty$, and construct such a cover via mirror descent. As applications of our framework, we show: $\bullet$ Matrix Spencer for Low-Rank Matrices. If the matrices satisfy $\|A_i\|_{\mathrm{op}} \leq 1$ and $\mathrm{rank}(A_i) \leq r$, we can efficiently find a coloring $x \in \{\pm 1\}^n$ with discrepancy $\|\sum_{i=1}^n x_i A_i \|_{\mathrm{op}} \lesssim \sqrt{n \log (\min(rm/n, r))}$. This improves upon the naive $O(\sqrt{n \log r})$ bound for random coloring and proves the matrix Spencer conjecture when $r m \leq n$. $\bullet$ Matrix Spencer for Block Diagonal Matrices. For block diagonal matrices with $\|A_i\|_{\mathrm{op}} \leq 1$ and block size $h$, we can efficiently find a coloring $x \in \{\pm 1\}^n$ with $\|\sum_{i=1}^n x_i A_i \|_{\mathrm{op}} \lesssim \sqrt{n \log (hm/n)}$. Using our proof, we reduce the matrix Spencer conjecture to the existence of a $O(\log(m/n))$ quantum relative entropy net on the spectraplex. $\bullet$ Matrix Discrepancy for Schatten Norms. We generalize our discrepancy bound for matrix Spencer to Schatten norms $2 \le p \leq q$. Given $\|A_i\|_{S_p} \leq 1$ and $\mathrm{rank}(A_i) \leq r$, we can efficiently find a partial coloring $x \in [-1,1]^n$ with $|\{i : |x_i| = 1\}| \ge n/2$ and $\|\sum_{i=1}^n x_i A_i\|_{S_q} \lesssim \sqrt{n \min(p, \log(rk))} \cdot k^{1/p-1/q}$, where $k := \min(1,m/n)$.