AI 中文总结
研究基于向量水平集熵估计，开发研究矩阵乘积\(\Pi U_V\)谱范数的方法，在给定\(k\)、\(r\)、\(p\)条件下，证明\(\|\Pi U_V\|\le C\sqrt{kp}\)，且该结果适用于负相关元素的随机模型。
AI 中文摘要
设\(\Pi\)为\(k\times n\)稀疏随机矩阵。对于固定的\(r\)维子空间\(V\subset{\mathbb R}^n\)，令\(U_V:{\mathbb R}^r\to{\mathbb R}^n\)表示从\({\mathbb R}^r\)到\(V\)的等距映射。乘积\(\Pi U_V\)是随机降维中的核心模型，主要通过迹和高斯比较不等式进行研究。在这项工作中，我们基于向量\(x\in V\)的水平集的熵估计，开发了一种方法来研究矩阵乘积\(\Pi U_V\)的谱范数。将该方法与现有估计相结合，我们得到如下结果。假设\(k\ge C\,r(\log\log r)^2\)且\(p\ge (\log k)/k\)。设\(\Pi\)为\(k\times n\)矩阵，其独立同分布的元素与乘积\(b\,\xi\)等分布，其中\(b\)是伯努利（\(p\)）随机变量，\(\xi\)均值为零，与\(b\)独立，且几乎必然满足\(|\xi|\le1\)。那么以高概率有\(\|\Pi U_V\|\le C\sqrt{kp}\)。对于具有负相关元素的其他随机模型，也有匹配的结果。
英文摘要
Let $Π$ be a $k\times n$ sparse random matrix. For a fixed $r$-dimensional subspace $V\subset{\mathbb R}^n$, let $U_V:{\mathbb R}^r\to{\mathbb R}^n$ denote an isometry from ${\mathbb R}^r$ onto $V$. The product $ΠU_V$ is a central model in randomized dimension reduction and has been studied primarily through trace and Gaussian comparison inequalities. In this work, we develop an approach to the spectral norm of the matrix product $ΠU_V$, based on entropy estimates for level sets of vectors $x\in V$. Combining the method with existing estimates, we show the following. Assume that \[ k\ge C\,r(\log\log r)^2,\qquad p\ge (\log k)/k. \] Let $Π$ be a $k\times n$ matrix with i.i.d. entries equidistributed with the product $b\,ξ$, where $b$ is a Bernoulli($p$) random variable and $ξ$ is mean-zero, independent of $b$, and satisfies $|ξ|\le1$ almost surely. Then with high probability \[ \|ΠU_V\|\le C\sqrt{kp}. \] Matching results hold for other random models with negatively associated entries.
