1W-MINDS, Nov. 20: Haotian Jiang (University of Chicago), Beck-Fiala and Komlos Bounds Beyond ... скачать в хорошем качестве

1W-MINDS, Nov. 20: Haotian Jiang (University of Chicago), Beck-Fiala and Komlos Bounds Beyond ...

Beck-Fiala and Komlos Bounds Beyond Banaszczyk The Beck-Fiala Conjecture asserts that any set system of n elements with degree k has combinatorial discrepancy O(√k). A substantial generalization is the Koml\'os Conjecture, which states that any m by n matrix with columns of unit Euclidean length has discrepancy O(1). In this talk, we describe an O'(log^1/4 n) bound for the Komlos problem, improving upon the O(log^1/2 n) bound due to Banaszczyk from 1998. We will also see how these ideas can be used to resolve the Beck-Fiala Conjecture for k \geq \log^2 n, and give a O'(k^1/2 + log^1/2 n) bound for smaller k, which improves upon Banaszczyk's O(k^1/2 log^1/2 n) bound. These results are based on a new technique of "Decoupling via Affine Spectral Independence" in designing rounding algorithms, which might also be useful in other contexts. This talk is based on joint work with Nikhil Bansal (University of Michigan).