Abstract. This technical report demonstrates theoretically and empirically that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results for OMP, which require O(m 2) measurements. The new results for OMP are comparable with recent results for another algorithm called Basis Pursuit (BP). The OMP algorithm is faster and easier to implement, which makes it an attractive alternative to BP for signal recovery problems. 1.

geometry of random polytopes

Abstract. We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n −1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum � k akXk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p. 1.

Abstract. Let n be a large integer and Mn be a random n by n matrix whose entries are i.i.d. Bernoulli random variables (each entry is±1 with probability 1/2). We show that the probability that Mn is singular is at most (3/4+o(1)) n, improving an earlier estimate of Kahn, Komlós and Szemerédi [11], as well as earlier work by the authors [17]. The key new ingredient is the applications of Freiman type inverse theorems and other tools from additive combinatorics. 1.

In this paper we consider four previously known parameters of sign matrices from a complexity-theoretic perspective. The main technical contributions are tight (or nearly tight) inequalities that we establish among these parameters. Several new open problems are raised as well. 1.

Abstract. Consider a random sum η1v1 +... + ηnvn, where η1,..., ηn are i.i.d. random signs and v1,..., vn are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as P(η1v1+...+ηnvn = 0) subject to various hypotheses on the v1,..., vn. In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman’s inverse theory) in additive combinatorics, which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the v1,..., vn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number. 1.

Abstract. This papers contains two results concerning random n×n Bernoulli matrices. First, we show that with probability tending to one the determinant has absolute value √ n!exp(O ( √ nln n)). Next, we prove a new upper bound.939 n on the probability that the matrix is singular. 1.

Abstract. Let A be an n × n matrix, whose entries are independent copies of a centered random variable satisfying the subgaussian tail estimate. We prove that the operator norm of A −1 does not exceed Cn 3/2 with probability close to 1. 1. Introduction. Let A be an n × n matrix, whose entries are independent identically distributed random variables. The spectral properties of such matrices, in particular invertibility, have been extensively studied (see, e.g. [M] and the survey [DS]). While A is almost surely invertible whenever its

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A system for private stream searching, introduced by Ostrovsky and Skeith [18], allows a client to provide an untrusted server with an encrypted search query. The server uses the query on a stream of documents and returns the matching documents to the client while learning nothing about the nature of the query. We present a new scheme for conducting private keyword search on streaming data which requires O(m) server to client communication complexity to return the content of the matching documents, where m is the size of the documents. The required storage on the server conducting the search is also O(m). The previous best scheme for private stream searching was shown to have O(m log m) communication and storage complexity. Our solution employs a novel construction in which the user reconstructs the matching files by solving a system of linear equations. This allows the matching documents to be stored in a compact buffer rather than relying on redundancies to avoid collisions in the storage buffer as in previous work. This technique requires a small amount of metadata to be returned in addition to the documents; for this the original scheme of Ostrovsky and Skeith may be employed with O(m log m) communication and storage complexity. We also present an alternative method for returning the necessary metadata based on a unique encrypted Bloom filter construction. This method requires O(m log(t/m)) communication and storage complexity, where t is the number of documents in the stream. The latter method results in much lower communication in most practical situations. In particular, if the number of matching documents is expected to be a fixed fraction of the stream length, the latter method results in the optimal O(m) overall communication and storage complexity with near optimal constant factors. In this paper we describe our scheme, prove it secure, analyze its asymptotic performance, and describe a number of extensions. We also provide an experimental analysis of its scalability in practice. Specifically, we consider its performance in the demanding scenario of providing a privacy preserving version of the Google News Alerts service.