Title: Efficient Statistics from Truncated and Dependent Samples
Abstract: Statistical estimation commonly assumes access to uncensored and independent data. We present recent work on efficient statistics from censored and dependent data. Our first result is an efficient algorithm for a problem, going back to Galton, Pearson, and Fisher, of estimating the parameters of a multivariate normal from truncated samples. Truncation is a strong type of censoring, which occurs when samples falling outside of some subset S of the support of the distribution are not observed, and their count in proportion to the observed samples is also not observed. Our second result is an efficient algorithm for linear and logistic regression in settings where the response variables are dependent, e.g. collected on a social network. Both methods are based on stochastic gradient descent, and use concentration and anticoncentration results for functions of dependent random variables.
(Based on joint works with Dikkala, Gouleakis, Panageas, Tzamos, and Zampetakis)
Title: Geodesic trees and Brownian watermelons: universality in models of local random growth
Abstract: An important technique for understanding a random system is to find a higher dimensional random system that enjoys an attractive and tractable structure and that has the system of interest as a marginal; and to analyse the new structure to make inferences about the original system. For example, the Airy_2 process is a random process mapping the real line to itself which is natural and significant because it offers, rigorously in certain examples and putatively in very many more, a scaled description at advanced time of a random interface whose growth is stimulated by local randomness and which is subject to restoring forces such as surface tension. The Airy_2 process may be embedded in a canonical way as the uppermost curve in a richer random object, the Airy line ensemble - an ordered system of random continuous curves. This richer object has an attractive probabilistic property not apparent in the Airy_2 process itself - it is, with suitable boundary conditions, an infinite system of mutually avoiding Brownian motions; and, as such, it enjoys a natural resampling property called the Brownian Gibbs property. The Brownian Gibbs property of the Airy line ensemble is a key probabilistic technique by which aspects of the concerned Kardar-Parisi-Zhang universality class of random growth models may be investigated. This talk will explain how, harnessed with limited but essential inputs of integrable origin, the property has recently been exploited to make very strong inferences regarding the locally Brownian nature of the Airy_2 process; about the scaled coalescence behaviour of geodesics in last passage percolation growth models; and about the structure of the scaled interface when these models are initiated from very general initial conditions.
Title: Generalization of TASEP in continuous inhomogeneous space
Abstract: We investigate a new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP).
One of the features of the particle systems we consider is the presence of spatial inhomogeneity which can lead to the formation of traffic jams.
For systems with special step-like initial data, we find explicit limit shapes, describe their hydrodynamic evolution, and obtain asymptotic fluctuation results which put our generalized TASEPs into the Kardar-Parisi-Zhang universality class. At a critical scaling around a traffic jam we observe deformations of the Tracy-Widom distribution and the extended Airy kernel.
The exact solvability and asymptotic behavior of generalizations of TASEP we study are powered by a new nontrivial connection to Schur measures and processes.
Based on joint work with Leonid Petrov and Axel Saenz.
Title: Survival and extinction of epidemics on random graphs with general degrees
Abstract: We establish the necessary and sufficient criterion for the contact process on Galton-Watson trees (resp. random graphs) to exhibit the phase of extinction (resp. short survival). We prove that the survival threshold $\lambda_1$ for a Galton-Watson tree is strictly positive if and only if its offspring distribution has an exponential tail, settling a conjecture by Huang and Durrett. On the random graph with degree distribution $D$, we show that if $D$ has an exponential tail, then for small enough $\lambda$ the contact process with the all-infected initial condition survives for polynomial time with high probability, while for large enough $\lambda$ it runs over exponential time with high probability. When $D$ is subexponential, the contact process typically displays long survival for any fixed $\lambda>0$. Joint work with Shankar Bhamidi, Danny Nam, and Allan Sly.
Title: Stochastic homogenization
Abstract: Stochastic homogenization is the PDE interpretation of quenched invariance principles for random walks in random environments. A crucial object is the corrector, which provides harmonic coordinates.
We show that large-scale fluctuations of a general solution to such an elliptic equation with random coefficients can be reduced to those of the corrector, which in turn are Gaussian, both to leading order. The arguments blend elliptic regularity theory and elementary Malliavin calculus.
This is joint work with M. Duerinckx and A. Gloria.
Title: Capacity lower bound for the Ising perceptron
Abstract: The perceptron is a toy model of a simple neural network that stores a collection of (randomly) given patterns. Its analysis reduces to a simple problem in high-dimensional geometry, namely, understanding the intersection of the cube (or sphere) with a collection of random half-spaces. Despite the simplicity of this model, its first-order asymptotic behavior is still not fully understood. I will describe what is known, and present some recent progress. This talk is based on joint work with Jian Ding.