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

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),...,A(n) is an element of R-mxm, a Gaussian measure lower bound of 2(-O (n)) for a scaling of the discrepancy body {x is an element of R-n : parallel to Sigma(n)(i=1) x(i)A(i) parallel to <= 1}. We show this is equivalent to covering its polar with 2(O (n)) translates of the cube 1/n B-infinity(n), and construct such a cover via mirror descent. As applications of our framework, we show: Matrix Spencer for Low-Rank Matrices. If the matrices satisfy parallel to A(i)parallel to(op) <= 1 and rank (A(i)) <= r, we can efficiently find a coloring x is an element of{+/- 1}(n) with discrepancy parallel to Sigma(n)(i=1) x(i)Lambda(i)parallel to(op) less than or similar to root nlog(min(rm/n,r)). This improves upon the naive O (root n log r) bound for random coloring and proves the matrix Spencer conjecture when rm <= n. Matrix Spencer for Block Diagonal Matrices. For block diagonal matrices with parallel to A(i)parallel to(op) <= 1 and block size h, we can efficiently find a coloring x is an element of{+/- 1}(n) with. parallel to Sigma(n)(i=1) x(i)A(i) parallel to(op) less than or similar to root nlog(hm/n). This bound was previously shown in [Levy, Ramadas and Rothvoss, IPCO 2017] under the assumption h <= root n, which we remove. 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. Matrix Discrepancy for Schatten Norms. We generalize our discrepancy bound for matrix Spencer to Schatten norms 2 <= p <= q. Given parallel to A(i)parallel to(Sp) less than or similar to root nmin(p, log(rk)) center dot k(1/p-1/q), = 1 where k := min(1, m/n). Our partial coloring bound is tight when m = Theta(root n). We also provide tight lower bounds of Omega(root n) for rank-1 matrix Spencer when m = n, and Omega(root min(m, n)) for S-2 -> S-infinity discrepancy, precluding a matrix version of the Komlos conjecture.