I will present an overview and some proofs from our work with Roman Vershynin.

We study n by n matrices A with i.i.d. entries. If the entries are also zero mean subgaussian, then the operator norm ||A|| ~ sqrt(n) with high probability, but for the distributions with heavier tails the norm can be significantly larger. We were motivated by the question: under what conditions the operator norm of a heavy-tailed matrix can be improved by modifying just a small fraction of its entries (a small sub-matrix of A)? We have shown that this happens exactly when the entries of A have zero mean and bounded variance.

I am going to discuss how enforcing the sqrt(n)-norm for the matrix can be a local or a global problem, depending on the moments of its entries. As parts of the proof, I am planning to talk about the relationships among various matrix norms, discretization of a random variable as a sum of independent Bernoullis, and Grothendieck-Pietsch factorization for matrices. Speaker(s): Elizaveta Rebrova (UM)