Title: Discrepancy Theory and Graph Sparsification

Advisors: Thomas Rothvoss and Shayan Oveis Gharan

Abstract: 

The Matrix Spencer Conjecture asks whether given n symmetric matrices in R^{n × n} with eigenvalues in [−1,1] one can always find signs so that their signed sum has singular values bounded by O(sqrt(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 a discrepancy algorithm due to Rothvoss can be used to sample a linear size sparsifer. In contrast to previous methods, we require only a logarithmic number of sampling phases.
Place: 
CSE2 (Gates Center) 371
When: 
Tuesday, October 15, 2019 - 13:30 to Thursday, March 28, 2024 - 17:24