Georgia Institute of Technology College of Sciences

The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from disciplines as diverse as genomics and social sciences. Motivated in part by previous work on this applied problem, this talk will present several new theoretical results concerning large average submatrices of an n x n Gaussian random matrix. We will begin by considering the average and joint distribution of the k x k submatrix having largest average value (the global maximum). We then turn our attention to submatrices with dominant row and column sums, which arise as the local maxima of a practical iterative search procedure for large average submatrices I will present a result characterizing the value and joint distribution of a local maximum, and show that a typical local maxima has an average value within a constant factor of the global maximum. In the last part of the talk I will describe several results concerning the *number* L_n(k) of k x k local maxima, including the asymptotic behavior of its mean and variance for fixed k and increasing n, and a central limit theorem for L_n(k) that is based on Stein's method for normal approximation. Joint work with Shankar Bhamidi (UNC) and Partha S. Dey (UIUC)

We analyze active learning algorithms, which only receive the classifications of examples when they ask for them, and traditional passive (PAC) learning algorithms, which receive classifications for all training examples, under log-concave and nearly log-concave distributions. By using an aggressive localization argument, we prove that active learning provides an exponential improvement over passive learning when learning homogeneous linear separators in these settings. Building on this, we then provide a computationally efficient algorithm with optimal sample complexity for passive learning in such settings. This provides the first bound for a polynomial-time algorithm that is tight for an interesting infinite class of hypothesis functions under a general class of data-distributions, and also characterizes the distribution-specific sample complexity for each distribution in the class. We also illustrate the power of localization for efficiently learning linear separators in two challenging noise models (malicious noise and agnostic setting) where we provide efficient algorithms with significantly better noise tolerance than previously known.

The classical Freidlin--Wentzell theory on small random perturbations of dynamical systems operates mainly at the level of large deviation estimates. In many cases it would be interesting and useful to supplement those with central limit theorem type results. We are able to describe a class of situations where a Gaussian scaling limit for the exit point of conditioned diffusions holds. Our main tools are Doob's h-transform and new gradient estimates for Hamilton--Jacobi equations. Joint work with Andrzej Swiech.

The Kardar-Parisi-Zhang(KPZ) equation is a non-linear stochastic partial di fferential equation proposed as the scaling limit for random growth models in physics. This equation is, in standard terms, ill posed and the notion of solution has attracted considerable attention in recent years. The purpose of this talk is two fold; on one side, an introduction to the KPZ equation and the so called KPZ universality classes is given. On the other side, we give recent results that generalize the notion of viscosity solutions from deterministic PDE to the stochastic case and apply these results to the KPZ equation. The main technical tool for this program to go through is a non-linear version of Feyman-Kac's formula that uses Doubly Backward Stochastic Differential Equations (Stochastic Differential Equations with times flowing backwards and forwards at the same time) as a basis for the representation.