Georgia Institute of Technology College of Sciences

Gabor systems, or collections of translations and modulations of a window function, are often used for time-frequency analysis of signals. The Balian-Low Theorem and its generalizations say that if a Gabor system obeys certain spanning and independence properties in L^2(R), then the window function of such a system cannot be well localized in both time and frequency. Recently, Shahaf Nitzan and Jan—Fredrik Olsen show that similar behavior extends to Gabor systems of finite length signals in l^2(Z_d). In this talk, I will discuss these finite dimensional results as well as recent extensions proven in collaboration with Josiah Park.

Logistic regression is arguably the most widely used and studied non-linear model in statistics. Classical maximum-likelihood theory based statistical inference is ubiquitous in this context. This theory hinges on well-known fundamental results—(1) the maximum-likelihood-estimate (MLE) is asymptotically unbiased and normally distributed, (2) its variability can be quantified via the inverse Fisher information, and (3) the likelihood-ratio-test (LRT) is asymptotically a Chi-Squared. In this talk, I will show that in the common modern setting where the number of features and the sample size are both large and comparable, classical results are far from accurate. In fact, (1) the MLE is biased, (2) its variability is far greater than classical results, and (3) the LRT is not distributed as a Chi-Square. Consequently, p-values obtained based on classical theory are completely invalid in high dimensions. In turn, I will propose a new theory that characterizes the asymptotic behavior of both the MLE and the LRT under some assumptions on the covariate distribution, in a high-dimensional setting. Empirical evidence demonstrates that this asymptotic theory provides accurate inference in finite samples. Practical implementation of these results necessitates the estimation of a single scalar, the overall signal strength, and I will propose a procedure for estimating this parameter precisely. This is based on joint work with Emmanuel Candes and Yuxin Chen.

Network data arises frequently in modern scientific applications. These networks often have specific characteristics such as edge sparsity, heavy-tailed degree distribution etc. Some broad challenges arising in the analysis of such datasets include (i) developing flexible, interpretable models for network datasets, (ii) testing for goodness of fit, (iii) provably recovering latent structure from such data.In this talk, we will discuss recent progress in addressing very specific instantiations of these challenges. In particular, we will1. Interpret the Caron-Fox model using notions of graph sub-sampling, 2. Study model misspecification due to rare, highly “influential” nodes, 3. Discuss recovery of community structure, given additional covariates.

In current convex optimization literature, there are significant gaps between algorithms that produce high accuracy (1+1/poly(n))-approximate solutions vs. algorithms that produce O(1)-approximate solutions for symmetrized special cases. This gap is reflected in the differences between interior point methods vs. (accelerated) gradient descent for regression problems, and between exact vs. approximate undirected max-flow. In this talk, I will discuss generalizations of a fundamental building block in numerical analysis, preconditioned iterative methods, to convex functions that include p-norms. This leads to algorithms that converge to high accuracy solutions by crudely solving a sequence of symmetric residual problems. I will then briefly describe several recent and ongoing projects, including p-norm regression using m^{1/3} linear system solves, p-norm flow in undirected unweighted graphs in almost-linear time, and further improvements to the dependence on p in the runtime.

In this talk, I will discuss the vanishing contact structure problem, which focuses on the asymptotic behavior of the viscosity solutions uε of Hamilton-Jacobi equation H (x, Du(x), ε u(x)) =c, as the factor ε tends to zero. This is a natural generalization of the vanishing discount problem. I will explain how to characterize the limit solution in terms of Peierls barrier functions and Mather measures from a dynamical viewpoint. This is a joint work with Hitoshi Ishii, Wei Cheng, and Kai Zhao.