In 2013, Marcus, Spielman, and Srivastava resolved the famous Kadison-Singer conjecture. It states that for $n$ independent random vectors $v_1,\cdots, v_n$ that have expected squared norm bounded by $\epsilon$ and are in the isotropic position in expectation, there is a positive probability that the determinant polynomial $\det(xI - \sum_{i=1}^n v_iv_i^\top)$ has roots bounded by $(1 + \sqrt{\epsilon})^2$. An interpretation of the Kadison-Singer theorem is that we can always find a partition of the vectors $v_1,\cdots,v_n$ 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: $\bullet$ The first one shows that the Kadison-Singer result requires a weaker assumption that the vectors have a bounded sum of hyperbolic norms. $\bullet$ 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.

翻译：2013年，Marcus、Spielman和Srivastava解决了著名的Kadison-Singer猜想。该猜想指出：对于$n$个独立随机向量$v_1,\cdots, v_n$，若其期望平方范数以$\epsilon$为界，且期望上处于各向同性位置，则存在正概率使得行列式多项式$\det(xI - \sum_{i=1}^n v_iv_i^\top)$的根以$(1 + \sqrt{\epsilon})^2$为界。Kadison-Singer定理的一种解读是：我们总能找到一种划分，将向量$v_1,\cdots,v_n$分成两组，使得它们在谱范数意义下具有低差异度（即依赖于行列式多项式）。本文针对更广泛的函数类——双曲多项式，给出了两项结果。此外，我们的结果适用于两种推广场景：$\bullet$ 第一项结果表明，Kadison-Singer结果仅需一个更弱的假设：向量具有有界的双曲范数之和。$\bullet$ 第二项结果将Kadison-Singer结果的分布假设放宽至强瑞利分布。据我们所知，先前的结果仅支持行列式多项式[Anari and Oveis Gharan'14, Kyng, Luh and Song'20]，尚不明确这些结果能否推广至更广泛的函数类。此外，我们还提出了一种亚指数时间算法用于构造我们的结果。