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.Comment: notation for the row product changed, references added, typos correcte

Invertibility of symmetric random matrices

We study n by n symmetric random matrices H, possibly discrete, with iid above-diagonal entries. We show that H is singular with probability at most exp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}). Furthermore, the spectrum of H is delocalized on the optimal scale o(n^{-1/2}). These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdos, Schlein and Yau.Comment: 53 pages. Minor corrections, changes in presentation. To appear in Random Structures and Algorithm

Random processes via the combinatorial dimension: introductory notes

This is an informal discussion on one of the basic problems in the theory of empirical processes, addressed in our preprint "Combinatorics of random processes and sections of convex bodies", which is available at ArXiV and from our web pages.Comment: 4 page