Many optimization problems arising in high-dimensional statistics decompose naturally into a sum of several terms, where the individual terms are relatively simple but the composite objective function can only be optimized with iterative algorithms. Specifically, we are interested in optimization problems of the form F(Kx) + G(x), where K is a fixed linear transformation, while F and G are functions that may be nonconvex and/or nondifferentiable. In particular, if either of the terms are nonconvex, existing alternating minimization techniques may…

We introduce a model for a high-dimensional random graph process and ask how "rich" the process has to be so that one finds atypical behavior. In particular, we study a natural process of Erdös-Rényi random graphs indexed by unit vectors in R^d . We investigate the deviations of the process with respect to three fundamental properties: clique number, chromatic number, and connectivity. The talk is based on joint work with Louigi Addario-Berry, Shankar Bhamidi, Sebastien Bubeck, Luc Devroye, and Roberto…

The number of publicly available gene expression datasets has been growing dramatically. Various methods had been proposed to predict gene co-expression by integrating the publicly available datasets. These methods assume that the genes in the query gene set are homogeneously correlated and consider no gene-specific correlation tendencies, no background intra-experimental correlations, and no quality variations of different experiments. We propose a two-step algorithm called CLIC (CLustering by Inferred Co-expression) based on a coherent Bayesian model to overcome these limitations. CLIC…

Professor Robins describes a novel approach to point and interval estimation of nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions. Higher order influence functions are higher order U-statistics. The approach applies equally to both n‾√ and non-n‾√ problems. It reproduces results previously obtained by the modern theory of non-parametric inference, produces many new n‾√ and non-n‾√ results, and opens up the ability to perform non-n‾√ inference in complex high dimensional models, such as models…

Missing data is a reality of empirical sciences and can rarely be prevented entirely. It is often assumed that incomplete data are missing completely at random (MCAR) or missing at random (MAR), When neither MCAR nor MAR, missingness is said to be Not MAR (NMAR). Under MAR, there are two main approaches to inference, likelihood/Bayesian inference, e.g. EM or MI, and semiparametric approaches such as Inverse probability weighting (IPW). In several important settings, likelihood based inferences suffer the difficulty of…

In game-theoretic formulations of prediction problems, a strategy makes a decision, observes an outcome and pays a loss. The aim is to minimize the regret, which is the amount by which the total loss incurred exceeds the total loss of the best decision in hindsight. This talk will focus on the minimax optimal strategy, which minimizes the regret, in three settings: prediction with log loss (a formulation of sequential probability density estimation that is closely related to sequential compression, coding,…

Principal components is a true workhorse of science and technology, applied everywhere from radio frequency signal processing to financial econometrics, genomics, and social network analysis. In this talk, I will review some of these applications and then describe the challenge posed by modern 'big data asymptotics' where there are roughly as many dimensions as observations; this setting has seemed in the past full of mysteries. Over the last ten years random matrix theory has developed a host of new tools…

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. This talk will present several new theoretical results concerning large average submatrices of an n x n Gaussian random matrix that are motivated in part by previous work on biomedical applications. We will begin by considering the average and distribution of the k x k submatrix having largest average value (the global maximum),…