Let K be an isotropic convex body in R n . Given ε > 0, how many independent points X i uniformly distributed on K are needed for the empirical covariance matrix to approximate the identity up to ε with overwhelming probability? Our paper answers this question from [12]. More precisely, let X ∈ R n be a centered random vector with a log-concave distribution and with the identity as covariance matrix. An example of such a vector X is a random point in an isotropic convex body. We show that for any ε > 0, there exists C(ε) > 0, such that if N ∼ C(ε) n and (X i ) i≤N are i.i.d. copies of, with probability larger than 1 − exp(−c √ n).
This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X 1 , . . . , ±X N ∈ R n , (N ≥ n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property (RIP) with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a sub-exponential tail inequality possess this property RIP with overwhelming probability. We show that such "sensing" matrices are valid for the exact reconstruction process of m-sparse vectors via ℓ 1 minimization with m ≤ Cn/ log 2 (cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K ⊂ R n is a convex body and X 1 , . . . , X N ∈ K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m ∼ n/ log 2 (cN/n).AMS Classification: primary 52A20, 94A12, 52B12, 46B09 secondary 15A52, 41A45, 94B75
Let 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 D n,d and M be its adjacency matrix. We show that M is invertible with probability at least 1where c, C are positive absolute constants. To this end, we establish a few properties of d-regular directed graphs. One of them, a Littlewood-Offord type anti-concentration property, is of independent interest. Let J be a subset of vertices of G with |J| ≈ n/d. Let δ i be the indicator of the event that the vertex i is connected to J and define δ = (δ 1 , δ 2 , ..., δ n ) ∈ {0, 1} n . Then for every v ∈ {0, 1} n the probability that δ = v is exponentially small. This property holds even if a part of the graph is "frozen." AMS 2010 Classification: 60C05, 60B20, 05C80, 15B52, 46B06.
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