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Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles
Let K be an isotropic convex body in Rn. Given e > 0, how many independent points Xi uniformly distributed on K are neededfor the empirical covariance matrix to approximate the identity up to e with
Smallest singular value of random matrices and geometry of random polytopes
Abstract We study the behaviour of the smallest singular value of a rectangular random matrix, i.e., matrix whose entries are independent random variables satisfying some additional conditions. We
Restricted Isometry Property of Matrices with Independent Columns and Neighborly Polytopes by Random Sampling
This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X1,…,±XN∈ℝn, (N≥n). We introduce a class of random sampling matrices
Orlicz norms of sequences of random variables
Let f i , i = 1,...,n, be copies of a random variable f and let N be an Orlicz function. We show that for every x ∈ R n the expectation E||(x i f i ) n i=1 || N is maximal (up to an absolute
Sharp bounds on the rate of convergence of the empirical covariance matrix
Let $X_1,..., X_N\in\R^n$ be independent centered random vectors with log-concave distribution and with the identity as covariance matrix. We show that with overwhelming probability at least $1 - 3
Euclidean projections of a p-convex body
In this paper we study Euclidean projections of a p-convex body in ℝn. Precisely, we prove that for any integer k satisfying ln n ≤ k ≤ n/2, there exists a projection of rank k with the distance to
John's Decomposition in the General Case and Applications
We give a description of an ane mapping T involving contact pairs of two general convex bodies K and L, when T(K) is in a position of maximal volume in L. This extends the classical John’s theorem
Averages of norms and quasi-norms
We compute the number of summands in q-averages of norms needed to approximate an Euclidean norm. It turns out that these numbers depend on the norm involved essentially only through the maximal
Adjacency matrices of random digraphs: singularity and anti-concentration
Let ${\mathcal D}_{n,d}$ be the set of all $d$-regular directed graphs on $n$ vertices. Let $G$ be a graph chosen uniformly at random from ${\mathcal D}_{n,d}$ and $M$ be its adjacency matrix. We
Condition number of a square matrix with i.i.d. columns drawn from a convex body
We study the smallest singular value of a square random matrix with i.i.d. columns drawn from an isotropic log-concave distribution. An important example is obtained by sampling vectors uniformly