A New Framework for Matrix Discrepancy: Partial Coloring Bounds via Mirror Descent
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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 ∈ℝ^m × m, a Gaussian measure lower bound of 2^-O(n) for a scaling of the discrepancy body {x ∈ℝ^n: ∑_i=1^n x_i A_i≤ 1}. We show this is equivalent to covering its polar with 2^O(n) translates of the cube 1/n B^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 A_i_op≤ 1 and rank(A_i) ≤ r, we can efficiently find a coloring x ∈{± 1}^n with discrepancy ∑_i=1^n x_i A_i _op≲√(n log (min(rm/n, r))). This improves upon the naive O(√(n log r)) bound for random coloring and proves the matrix Spencer conjecture when r m ≤ n. ∙ Matrix Spencer for Block Diagonal Matrices. For block diagonal matrices with A_i_op≤ 1 and block size h, we can efficiently find a coloring x ∈{± 1}^n with ∑_i=1^n x_i A_i _op≲√(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. ∙ Matrix Discrepancy for Schatten Norms. We generalize our discrepancy bound for matrix Spencer to Schatten norms 2 ≤ p ≤ q. Given A_i_S_p≤ 1 and rank(A_i) ≤ r, we can efficiently find a partial coloring x ∈ [-1,1]^n with |{i : |x_i| = 1}| ≥ n/2 and ∑_i=1^n x_i A_i_S_q≲√(n min(p, log(rk)))· k^1/p-1/q, where k := min(1,m/n).
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