Georgia Institute of Technology College of Sciences

Loosely speaking, the maximum likelihood threshold of a statistical model is the fewest number of data points needed to fit the model using maximum likelihood estimation. In this talk, I will discuss combinatorial and algebraic-geometric approaches to studying this poorly understood quantity for a certain class of Gaussian models. This is based on joint work with Sean Dewar, Steven Gortler, Tony Nixon, Meera Sitharam, and Louis Theran

Given a Legendrian knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We proved that the LOSS invariant is natural under the positive contact surgery. In this talk I will review some background and definition, try to get the ideal of the proof and talk about the application which is about distinguishing Legendrian and Transverse knot.

Estimating the fixed-point of a contractive operator from empirical data is a fundamental computational and statistical task. In many practical applications including dynamic programming, the relevant norm is not induced by an inner product structure, which hinders existing techniques for analysis. In this talk, I will present recent advances in stochastic approximation methods for fixed-point equations in Banach spaces. Among other results, we discuss a novel variance-reduced stochastic approximation scheme, and establish its non-asymptotic error bounds. In contrast to worst-case guarantees, our bounds are instance-dependent, and achieve the optimal covariance structure in central limit theorems non-asymptotically.
Joint works with Koulik Khamaru, Martin Wainwright, Peter Bartlett, and Michael Jordan.