Linear Size Sparsifier and the Geometry of the Operator Norm Ball

Presented on October 7, 2019 at 11:00 a.m. in the Klaus Advanced Computing Building, Room 1116E.Thomas Rothvoss holds a joint position as an Associate Professor in the Math and CSE departments at the University of Washington. His research interests are discrete optimization, linear/integer programming and theoretical computer science.Runtime: 55:21 minutesThe Matrix Spencer Conjecture asks whether given n symmetric matrices in ℝn×n with eigenvalues in [−1,1] one can always find signs so that their signed sum has singular values bounded by O(n‾√). The standard approach in discrepancy requires proving that the convex body of all good fractional signings is large enough. However, this question has remained wide open due to the lack of tools to certify measure lower bounds for rather small non-polyhedral convex sets. A seminal result by Batson, Spielman and Srivastava from 2008 shows that any undirected graph admits a linear size spectral sparsifier. Again, one can define a convex body of all good fractional signings. We can indeed prove that this body is close to most of the Gaussian measure. This implies that a discrepancy algorithm by the second author can be used to sample a linear size sparsifer. In contrast to previous methods, we require only a logarithmic number of sampling phases. This is joint work with Victor Reis