Georgia Institute of Technology College of Sciences

Spectral algorithms are a powerful method for detecting low-rank structure in dense random matrices and random graphs. However, in certain problems involving sparse random graphs with bounded average vertex degree, a naive spectral analysis of the graph adjacency matrix fails to detect this structure. In this talk, I will discuss a semidefinite programming (SDP) approach to address this problem, which may be viewed both as imposing a delocalization constraint on the maximum eigenvalue problem and as a natural convex relaxation of minimum graph bisection. I will discuss probabilistic results that bound the value of this SDP for sparse Erdos-Renyi random graphs with fixed average vertex degree, as well as an extension of the lower bound to the two-group stochastic block model. Our upper bound uses a dual witness construction that is related to the non-backtracking matrix of the graph. Our lower bounds analyze the behavior of local algorithms, and in particular imply that such algorithms can approximately solve the SDP in the Erdos-Renyi setting. This is joint work with Andrea Montanari.

Estimation of the covariance matrix has attracted significant attention of the statistical research community over the years, partially due to important applications such as Principal Component Analysis. However, frequently used empirical covariance estimator (and its modifications) is very sensitive to outliers, or ``atypical’’ points in the sample. As P. Huber wrote in 1964, “...This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): what happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance…” Motivated by Tukey's question, we develop a new estimator of the (element-wise) mean of a random matrix, which includes covariance estimation problem as a special case. Assuming that the entries of a matrix possess only finite second moment, this new estimator admits sub-Gaussian or sub-exponential concentration around the unknown mean in the operator norm. We will present extensions of our approach to matrix-valued U-statistics, as well as applications such as the matrix completion problem. Part of the talk will be based on a joint work with Xiaohan Wei.

As a general fact, directed polymers in random environment are localized in the so called strong disorder phase. In this talk, based on a joint with Francis Comets, we will consider the exactly solvable model with log gamma environment,introduced recently by Seppalainen. For the stationary model and the point to line version, the localization can be expressed as the trapping of the endpoint in a potential given by an independent random walk.

Motivated by the investigation on the dependence on ``epsilon" in the Dvoretzky's theorem, I will show some refinements of the classical concentration of measure for convex functions. Applications to convexity will be presented if time permits. The talk will be based on joint works with Peter Pivovarov and Petros Valettas.