Row products of random matrices

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.