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- W4226428014 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 $k := min(1,m/n)$." @default.
- W4226428014 created "2022-05-05" @default.
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- W4226428014 date "2021-11-04" @default.
- W4226428014 modified "2023-10-16" @default.
- W4226428014 title "A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent" @default.
- W4226428014 doi "https://doi.org/10.48550/arxiv.2111.03171" @default.
- W4226428014 hasPublicationYear "2021" @default.
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