School of Mathematics

We propose a new hierarchy of semidefinite programming relaxations for inference problems. As test cases, we consider the problem of community detection in block models. The vertices are partitioned into k communities, and a graph is sampled...

The factorization of Fourier coefficients of automorphic forms plays an important role in a wide range of topics, from the study of L-functions to the interpretation of scattering amplitudes in string theory. In this talk I will present a transfer...

Much real-world data is sampled at irregular intervals, but most time series models require regularly-sampled data. Continuous-time models address this problem, but until now only deterministic (ODE) models or linear-Gaussian models were efficiently...

In 1965, M. Kac proved that discs were uniquely determined by their Dirichlet (or, Neumann) spectra. Until recently, disks were the only smooth plane domains known to be determined by their eigenvalues. Recently, H. Hezari and I proved that ellipses...

We investigate the approximability of the following optimization problem, whose input is an n-by-n matrix A and an origin symmetric convex set C that is given by a membership oracle: "Maximize the quadratic form as x ranges over C." This is a rich...

A weighted graph H is a sparsifier of a graph G if H has much fewer edges than G and, in an appropriate technical sense, H "approximates" G. Sparsifiers are useful as compressed representations of graphs and to speed up certain graph algorithms. In...

Given a smooth knot K in the 3-sphere, a classic question in knot theory is: What surfaces in the 4-ball have boundary equal to K? One can also consider immersed surfaces and ask a “geography” question: What combinations of genus and double points...

Modern deep generative models like GANs, VAEs and invertible flows are showing amazing results on modeling high-dimensional distributions, especially for images. We will show how they can be used to solve inverse problems by generalizing compressed...

There has been tremendous progress in designing accurate black-box prediction methods (boosting, random forests, bagging, neural nets, etc.) but for deployment in the real world, it is useful to quantify uncertainty beyond making point-predictions...

This talk aims to summarize a project I was involved in during the past 5 years, with the hope of explaining our most complete understanding so far, as well as challenges and open problems. The main messages of this project are summarized below; I...