Zero-One Rounding of Singular Vectors

Given a matrix A, it can be shown that there is a vector z ∈ 0, 1 for which |Az|/|Z| ≥ |A|2/C log(n) (a0/1 sum of columns of A which witnesses its large spectral norm) for instance by discretizing the top singular vector of A and taking a dyadic expansion. We give a simple geometric proof of this fact by relating it to the Euclidean diameter of a polytope, and obtain a sharp bound of |Az| ≥ 2|A|2/ √ ln(n). We then show that it is possible to extract k orthogonal 0, 1 vectors z1 . . . zk such that |Azi|/|zi| ≥ σk(A)/ √ k lnn, which is tight up to a √ log k factor, and mention applications to approximating matrices by sums of rectangles. Joint work with Amit Deshpande and Ravi Kannan.