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

Shiri Artstein-Avidan·Omer Friedland·Vitali Milman·Sasha Sodin
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Abstract

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.

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Shiri Artstein-Avidan, Omer Friedland, Vitali Milman, Sasha Sodin. 2006-03-15. Polynomial bounds for large Bernoulli sections of $\ell_1^N$. https://arxiv.org/abs/math/0601369

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