On the interval of fluctuation of the singular values of random matrices

Abstract : Let A be a matrix whose columns X 1 ,. .. , X N are independent random vectors in R n. Assume that the tails of the 1-dimensional marginals decay as P(| i , a | ≥ t) ≤ t −p uniformly in a ∈ S n−1 and i ≤ N. Then for p > 4 we prove that with high probability A/ √ n has the Restricted Isometry Property (RIP) provided that Eu-clidean norms |X i | are concentrated around √ n. We also show that
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Olivier Guédon, Alexander Litvak, Alain Pajor, Nicole Tomczak-Jaegermann. On the interval of fluctuation of the singular values of random matrices. J. Eur. Math. Soc. (JEMS) 6, 2017, 19 (5), pp.1469-1505. ⟨hal-01262618⟩

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