On generic chaining and the smallest singular value of random matrices with heavy tails

We present a very general chaining method which allows one to control the supremum of the empirical process suph∈H |N−1 ∑N i=1 h (Xi)− Eh2| in rather general situations. We use this method to establish two main results. First, a quantitative (non asymptotic) version of the classical Bai-Yin Theorem on the singular values of a random matrix with i.i.d entries that have heavy tails, and second, a sharp estimate on the quadratic empirical process when H = {〈t, ·〉 : t ∈ T}, T ⊂ R and μ is an isotropic, unconditional, log-concave measure.