2026-07-25T12:07:37Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17132 2024-03-06T10:59:02Z ddc:004 open_access

Hyperbolic Concentration, Anti-Concentration, and Discrepancy Song, Zhao Zhang, Ruizhe Hyperbolic polynomial Chernoff bound Concentration Discrepancy theory Anti-concentration 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(√{n log n}) with high probability, while the famous result by Spencer [Spe85] shows that there exists an O(√n) discrepancy solution. The study of hyperbolic polynomials dates back to the early 20th century when used to solve PDEs by Gårding [Går59]. 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 [Brä18] 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 which is a special case of hyperbolic polynomials. 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Zhao Song and Ruizhe Zhang 2022 Is Part Of LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX/RANDOM.2022.10 urn:nbn:de:0030-drops-171324 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.10 eng https://creativecommons.org/licenses/by/4.0/legalcode