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

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

Small ball probabilities for linear images of high dimensional distributions

We study concentration properties of random vectors of the form $AX$, where $X = (X_1, ..., X_n)$ has independent coordinates and $A$ is a given matrix. We show that the distribution of $AX$ is well spread in space whenever the distributions of $X_i$ are well spread on the line. Specifically, assume that the probability that $X_i$ falls in any given interval of length $T$ is at most $p$. Then the probability that $AX$ falls in any given ball of radius $T \|A\|_{HS}$ is at most $(Cp)^{0.9 r(A)}$, where $r(A)$ denotes the stable rank of $A$ and $C$ is an absolute constant.Comment: 18 pages. A statement of Rogozin's theorem is added. Small corrections are mad

L_p moments of random vectors via majorizing measures

For a random vector X in R^n, we obtain bounds on the size of a sample, for which the empirical p-th moments of linear functionals are close to the exact ones uniformly on an n-dimensional convex body K. We prove an estimate for a general random vector and apply it to several problems arising in geometric functional analysis. In particular, we find a short Lewis type decomposition for any finite dimensional subspace of L_p. We also prove that for an isotropic log-concave random vector, we only need about n^{p/2} \log n sample points so that the empirical p-th moments of the linear functionals are almost isometrically the same as the exact ones. We obtain a concentration estimate for the empirical moments. The main ingredient of the proof is the construction of an appropriate majorizing measure to bound a certain Gaussian process.Comment: 32 pages, to appear in Advances in Mathematic

Geometric approach to error correcting codes and reconstruction of signals

We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way that x can be decoded correctly when any r letters of y are corrupted. We prove that most linear orthogonal transformations Q from R^n into R^m form efficient and robust robust error correcting codes over reals. The decoder (which corrects the corrupted components of y) is the metric projection onto the range of Q in the L_1 norm. An equivalent problem arises in signal processing: how to reconstruct a signal that belongs to a small class from few linear measurements? We prove that for most sets of Gaussian measurements, all signals of small support can be exactly reconstructed by the L_1 norm minimization. This is a substantial improvement of recent results of Donoho and of Candes and Tao. An equivalent problem in combinatorial geometry is the existence of a polytope with fixed number of facets and maximal number of lower-dimensional facets. We prove that most sections of the cube form such polytopes.Comment: 17 pages, 3 figure