Row products of random matrices

TL;DR

This paper investigates the spectral properties of row products of independent random matrices and their applications in privacy-preserving data release, revealing that these matrices behave similarly to matrices with independent entries under certain conditions.

Contribution

It introduces the analysis of spectral properties of row products of random matrices and applies these results to improve privacy-preserving data release methods.

Findings

Largest and smallest singular values are of the same order when n is much smaller than d^K.

Spectral properties of row product matrices resemble those of matrices with independent entries.

Derived bounds on noise addition for privacy protection in data release.

Abstract

We define the row product of K matrices of size d by n as a matrix of size d^K by n, whose row are entry-wise products of rows of these matrices. This construction arises in certain computer science problems. We study the question, to which extent the spectral and geometric properties of the row product of independent random matrices resemble those properties for a d^K by n matrix with independent random entries. In particular, we show that the largest and the smallest singular values of these matrices are of the same order, as long as n is significantly smaller than d^K. We also consider a problem of privately releasing the summary information about a database, and use the previous results to obtain a bound for the minimal amount of noise, which has to be added to the released data to avoid a privacy breach.