Zhang, Ruizhe

Zhang, Xinzhi

English

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

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A Hyperbolic Extension of Kadison-Singer Type Results

https://drops.dagstuhl.de/opus/volltexte/2023/18160/pdf/LIPIcs-ICALP-2023-108.pdf

A Hyperbolic Extension of Kadison-Singer Type Results

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