Level-set entropy and sparse randomized embeddings

2026-07-24

AI Summary

This study addresses the theoretical foundations of randomized dimensionality reduction by establishing sharp spectral norm bounds for the product of sparse random matrices and low-dimensional subspace embeddings. Departing from conventional approaches based on trace methods or Gaussian comparison inequalities, the work introduces a novel entropy-based analysis of vector level sets to achieve a refined understanding of the spectral properties of sparse random embeddings. By integrating models of negatively associated random variables with subspace isometry theory, the paper proves that, under the conditions \(k \geq C r(\log\log r)^2\) and \(p \geq (\log k)/k\), the spectral norm satisfies \(\|\Pi U_V\| \leq C\sqrt{kp}\) with high probability. This result is further extended to a broader class of negatively associated random matrix ensembles.

Abstract

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.

Problem

• level-set entropy
• sparse randomized embeddings
• spectral norm
• random matrix
• dimension reduction

Innovation

• level-set entropy
• sparse randomized embeddings
• spectral norm
• randomized dimension reduction
• negatively associated entries