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

Olivier Guédon (1) , Alexander Litvak (2) , Alain Pajor (1) , Nicole Tomczak-Jaegermann (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 University of Alberta

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

Domaines: Analyse fonctionnelle [math.FA]; Probabilités [math.PR]

Olivier Guédon, Alexander Litvak, Alain Pajor, Nicole Tomczak-Jaegermann. On the interval of fluctuation of the singular values of random matrices. Journal of the European Mathematical Society, 2017, 19 (5), pp.1469-1505. ⟨10.4171/JEMS/697⟩. ⟨hal-01262618⟩