Polynomial bounds for large Bernoulli sections of $\ell_1^N$

Artstein-Avidan, Shiri; Friedland, Omer; Milman, Vitali; Sodin, Sasha

Polynomial bounds for large Bernoulli sections of $\ell_1^N$

Authors

Artstein-Avidan, Shiri

Friedland, Omer

Milman, Vitali

Sodin, Sasha

Description

We prove a quantitative version of the bound on the smallest singular value of a Bernoulli covariance matrix (due to Bai and Yin). Then we use this bound, together with several recent developments, to show that the distance from a random (1-delta) n - dimensional section of ell_1^n, realised as an image of a sign matrix, to an Euclidean ball is polynomial in 1/delta (and independent of n), with high probability.
22 pages