Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling
Résumé
This paper considers compressed sensing matrices and neighbor- liness of a centrally symmetric convex polytope generated by vectors ±X1,...,±XN ∈ Rn, (N ≥ n). We introduce a class of random sam- pling matrices and show that they satisfy a restricted isometry prop- erty (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" matri- ces are valid for the exact reconstruction process of m-sparse vectors via l1 minimization with m ≤ Cn/log2(cN/n). The class of sam- pling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K ⊂ Rn is a convex body and X1,...,XN ∈ K are i.i.d. random vectors uniformly distributed on K, then, with over- whelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m ∼ n/ log2(cN/n).