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Track A: Algorithms, Complexity and Games
A Hyperbolic Extension of Kadison-Singer Type Results

Authors: Ruizhe Zhang and Xinzhi Zhang

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)


Abstract
In 2013, Marcus, Spielman, and Srivastava resolved the famous Kadison-Singer conjecture. It states that for n independent random vectors v_1,⋯, v_n that have expected squared norm bounded by ε and are in the isotropic position in expectation, there is a positive probability that the determinant polynomial det(xI - ∑_{i=1}^n v_i v_i^⊤) has roots bounded by (1 + √ε)². An interpretation of the Kadison-Singer theorem is that we can always find a partition of the vectors v_1,⋯,v_n into two sets with a low discrepancy in terms of the spectral norm (in other words, rely on the determinant polynomial). In this paper, we provide two results for a broader class of polynomials, the hyperbolic polynomials. Furthermore, our results are in two generalized settings: - The first one shows that the Kadison-Singer result requires a weaker assumption that the vectors have a bounded sum of hyperbolic norms. - The second one relaxes the Kadison-Singer result’s distribution assumption to the Strongly Rayleigh distribution. To the best of our knowledge, the previous results only support determinant polynomials [Anari and Oveis Gharan'14, Kyng, Luh and Song'20]. It is unclear whether they can be generalized to a broader class of polynomials. In addition, we also provide a sub-exponential time algorithm for constructing our results.

Cite as

Ruizhe Zhang and Xinzhi Zhang. A Hyperbolic Extension of Kadison-Singer Type Results. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 108:1-108:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{zhang_et_al:LIPIcs.ICALP.2023.108,
  author =	{Zhang, Ruizhe and Zhang, Xinzhi},
  title =	{{A Hyperbolic Extension of Kadison-Singer Type Results}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{108:1--108:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.108},
  URN =		{urn:nbn:de:0030-drops-181606},
  doi =		{10.4230/LIPIcs.ICALP.2023.108},
  annote =	{Keywords: Kadison-Singer conjecture, Hyperbolic polynomials, Strongly-Rayleigh distributions, Interlacing polynomials}
}
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