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

Abstract

We present a very general chaining method which allows one to control the supremum of the empirical process sup h∈H |N −1 N i=1 h 2 (X i )− Eh 2 | 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 n and µ is an isotropic, unconditional, log-concave measure.

Sample Covariance Matrices of Heavy-Tailed Distributions

“…Let us remark that estimates for quadratic forms similar to the ones above, appeared in literature before. In particular, we refer to a work of J. Bourgain [7], which deals with approximating covariance matrices of log-concave distributions (see also [1]), as well as papers [19,20] where a "chaining" argument was employed for dealing with heavy-tailed distributions (see [10] for further development of the technique). Unlike the above computations, the next lemma is a new addition to the arguments employed in [20,10].…”

On Multiplier Processes Under Weak Moment Assumptions

“…There are other generic situations in whichΛ s 0 ,u (F ) may be controlled using the geometry of F (for example [13,9] when F is a class of linear functionals on R n and X is an unconditional, log-concave random vector). However, there is no satisfactory theory that describesΛ s 0 ,u (F ) for an arbitrary class F ; such results are highly nontrivial.…”

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

“…Of course, in view of Bai-Yin theorem, it is a natural question whether one can replace the function φ(t) = e t /2 by the function φ(t) = e t α /2 with α ∈ (0, 1) or φ(t) = t p , for p ≥ 4. The first attempt in this direction was done in [34], where the bound S ≤ C(p, K)(n/N) 1/2−2/p (ln ln n) 2 was obtained for every p > 4 provided that M ≤ K √ n. Clearly, ln ln n is a "parasitic" term, which, in particular, does not allow to solve the KLS problem with N proportional to n. This problem was solved in [24,31] under strong assumptions and in particular when M ≤ K √ n and X has i.i.d. coordinates with bounded p-th moment with p > 4.…”