Georgia Institute of Technology College of Sciences

High-dimensional statistics is the basis for analyzing large and complex data sets that are generated by cutting-edge technologies in genetics, neuroscience, astronomy, and many other fields. However, Lasso, Ridge Regression, Graphical Lasso, and other standard methods in high-dimensional statistics depend on tuning parameters that are difficult to calibrate in practice. In this talk, I present two novel approaches to overcome this difficulty. My first approach is based on a novel testing scheme that is inspired by Lepski’s idea for bandwidth selection in non-parametric statistics. This approach provides tuning parameter calibration for estimation and prediction with the Lasso and other standard methods and is to date the only way to ensure high performance, fast computations, and optimal finite sample guarantees. My second approach is based on the minimization of an objective function that avoids tuning parameters altogether. This approach provides accurate variable selection in regression settings and, additionally, opens up new possibilities for the estimation of gene regulation networks, microbial ecosystems, and many other network structures.

This talk will be about random lozenge tilings of a class of planar domains which I like to call "sawtooth domains." The basic question is: what does a uniformly random lozenge tiling of a large sawtooth domain look like? At the first order of randomness, a remarkable form of the law of large numbers emerges: the height function of the tiling converges to a deterministic "limit shape." My talk is about the next order of randomness, where one wants to analyze the fluctuations of tiles around their eventual positions in the limit shape. Quite remarkably, this analytic problem can be solved in an essentially combinatorial way, using a desymmetrized version of the double Hurwitz numbers from enumerative algebraic geometry.

In first-passage percolation (FPP), one places random non-negative weights on the edges of a graph and considers the induced weighted graph metric. Of particular interest is the case where the graph is Z^d, the standard d-dimensional cubic lattice, and many of the questions involve a comparison between the asymptotics of the random metric and the standard Euclidean one. In this talk, I will survey some of my recent work on the order of fluctuations of the metric, focusing on (a) lower bounds for the expected distance and (b) our recent sublinear bound for the variance for edge-weight distributions that have 2+log moments, with corresponding concentration results. This second work addresses a question posed by Benjamini-Kalai-Schramm in their celebrated 2003 paper, where such a bound was proved for only Bernoulli weights using hypercontractivity. Our techniques draw heavily on entropy methods from concentration of measure.

Spin glasses are disordered spin systems originated from the desire of understanding the strange magnetic behaviors of certain alloys in physics. As mathematical objects, they are often cited as examples of complex systems and have provided several fascinating structures and conjectures. This talk will be focused on one of the famous mean-field spin glasses, the Sherrington-Kirkpatrick model. We will present results on the conjectured properties of the Parisi measure including its uniqueness and quantitative behaviors. This is based on joint works with A. Auffinger.