Hyperbolic Concentration, Anti-concentration, and Discrepancy

Abstract

Chernoff bound is a fundamental tool in theoretical computer science. It has been extensively used in randomized algorithm design and stochastic type analysis. Discrepancy theory, which deals with finding a bi-coloring of a set system such that the coloring of each set is balanced, has a huge number of applications in approximation algorithms design. Chernoff bound [Che52] implies that a random bi-coloring of any set system with $n$ sets and $n$ elements will have discrepancy $O(\sqrt{n \log n})$ with high probability, while the famous result by Spencer [Spe85] shows that there exists an $O(\sqrt{n})$ discrepancy solution. The study of hyperbolic polynomials dates back to the early 20th century when used to solve PDEs by G{\aa}rding [G{\aa}r59]. In recent years, more applications are found in control theory, optimization, real algebraic geometry, and so on. In particular, the breakthrough result by Marcus, Spielman, and Srivastava [MSS15] uses the theory of hyperbolic polynomials to prove the Kadison-Singer conjecture [KS59], which is closely related to discrepancy theory. In this paper, we present a list of new results for hyperbolic polynomials: * We show two nearly optimal hyperbolic Chernoff bounds: one for Rademacher sum of arbitrary vectors and another for random vectors in the hyperbolic cone. * We show a hyperbolic anti-concentration bound. * We generalize the hyperbolic Kadison-Singer theorem [Br\"a18] for vectors in sub-isotropic position, and prove a hyperbolic Spencer theorem for any constant hyperbolic rank vectors. The classical matrix Chernoff and discrepancy results are based on determinant polynomial. To the best of our knowledge, this paper is the first work that shows either concentration or anti-concentration results for hyperbolic polynomials. We hope our findings provide more insights into hyperbolic and discrepancy theories.