Georgia Institute of Technology College of Sciences

I will talk about the structure of large square random matrices with centered i.i.d. heavy-tailed entries (only two finite moments are assumed). In our previous work with R. Vershynin we have shown that the operator norm of such matrix A can be reduced to the optimal sqrt(n)-order with high probability by zeroing out a small submatrix of A, but did not describe the structure of this "bad" submatrix, nor provide a constructive way to find it. Now we can give a very simple description of this small "bad" subset: it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms to bring its operator norm to the almost optimal sqrt(loglog(n)*n)-order, under additional assumption that the entries of A are symmetrically distributed. As a corollary, one can also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same regularization.
Im am planning to discuss some details of the proof, the main component of which is the development of techniques that extend constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.

We identify principal component analysis (PCA) as an empirical risk minimization problem with respect to the reconstruction error and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. We also discuss how our results can be transferred to the subspace distance and, for instance, how our approach leads to a sharp $\sin \Theta$ theorem for empirical covariance operators. The proof is based on a novel contraction property, contrasting previous spectral perturbation approaches. This talk is based on joint works with Markus Reiß and Moritz Jirak.

Wiener-Hopf factorization (WHf) encompasses several important results in probability and stochastic processes, as well as in operator theory. The importance of the WHf stems not only from its theoretical appeal, manifested, in part, through probabilistic interpretation of analytical results, but also from its practical applications in a wide range of fields, such as fluctuation theory, insurance and finance. The various existing forms of the WHf for Markov chains, strong Markov processes, Levy processes, and Markov additive process, have been obtained only in the time-homogeneous case. However, there are abundant real life dynamical systems that are modeled in terms of time-inhomogenous processes, and yet the corresponding Wiener-Hopf factorization theory is not available for this important class of models. In this talk, I will first provide a survey on the development of Wiener-Hopf factorization for time-homogeneous Markov chains, Levy processes, and Markov additive processes. Then, I will discuss our recent work on WHf for time-inhomogensous Markov chains. To the best of our knowledge, this study is the first attempt to investigate the WHf for time-inhomogeneous Markov processes.