Seminars and Colloquia by Series

Prevalence of heavy-tailed distributions in systems with multiple scales: insights through stochastic averaging

Series
Stochastics Seminar
Time
Thursday, November 29, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Rachel Kuske – School of Mathematics, GaTech
Heavy tailed distributions have been shown to be consistent with data in a variety of systems with multiple time scales. Recently, increasing attention has appeared in different phenomena related to climate. For example, correlated additive and multiplicative (CAM) Gaussian noise, with infinite variance or heavy tails in certain parameter regimes, has received increased attention in the context of atmosphere and ocean dynamics. We discuss how CAM noise can appear generically in many reduced models. Then we show how reduced models for systems driven by fast linear CAM noise processes can be connected with the stochastic averaging for multiple scales systems driven by alpha-stable processes. We identify the conditions under which the approximation of a CAM noise process is valid in the averaged system, and illustrate methods using effectively equivalent fast, infinite-variance processes. These applications motivate new stochastic averaging results for systems with fast processes driven by heavy-tailed noise. We develop these results for the case of alpha-stable noise, and discuss open problems for identifying appropriate heavy tailed distributions for these multiple scale systems. This is joint work with Prof. Adam Monahan (U Victoria) and Dr. Will Thompson (UBC/NMi Metrology and Gaming).

Random walks with relocations and memory through random recursive trees

Series
Stochastics Seminar
Time
Thursday, November 15, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Geronimo Uribe – UNAM –
(Based on joint work with Cécile Mailler)Consider a stochastic process that behaves as a d-dimensional simple and symmetric random walk, except that, with a certain fixed probability, at each step, it chooses instead to jump to a given site with probability proportional to the time it has already spent there. This process has been analyzed in the physics literature under the name "random walk with preferential relocations", where it is argued that the position of the walker after n steps, scaled by log(n), converges to a Gaussian random variable; because of the log spatial scaling, the process is said to undergo a "slow diffusion". We generalize this model by allowing the underlying random walk to be any Markov process and the random run-lengths (time between two relocations) to be i.i.d.-distributed. We also allow the memory of the walker to fade with time, meaning that when a relocations occurs, the walker is more likely to go back to a place it has visited more recently. We prove rigorously the central limit theorem described above by associating to the process a growing family of vertex-weighted random recursive trees and a Markov chain indexed by this tree. The spatial scaling of our relocated random walk is related to the height of a typical vertex in the random tree. This typical height can range from doubly-logarithmic to logarithmic or even a power of the number of nodes of the tree, depending on the form of the memory.

Stabilization of Diffusion Limited Aggregation in a Wedge

Series
Stochastics Seminar
Time
Thursday, October 25, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Eviatar Procaccia – Texas A&M –
We prove a discrete Beurling estimate for the harmonic measure in a wedge in $\mathbf{Z}^2$, and use it to show that Diffusion Limited Aggregation (DLA) in a wedge of angle smaller than $\pi/4$ stabilizes. This allows to consider the infinite DLA as a finite time growth process and questions about the number of arms, growth and dimension. I will present some conjectures and open problems. This is joint work with Ron Rosenthal (Technion) and Yuan Zhang (Pekin University).

Lectures on Combinatorial Statistics: 2

Series
Stochastics Seminar
Time
Thursday, October 18, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Gabor Lugosi – Pompeu Fabra University, Barcelona –
In these lectures we discuss some statistical problems with an interesting combinatorial structure behind. We start by reviewing the "hidden clique" problem, a simple prototypical example with a surprisingly rich structure. We also discuss various "combinatorial" testing problems and their connections to high-dimensional random geometric graphs. Time permitting, we study the problem of estimating the mean of a random variable

Majority vote model on the 3-regular tree

Series
Stochastics Seminar
Time
Thursday, October 11, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael Damron – Georgia Institute of Technology –
In the continuous-time majority vote model, each vertex of a graph is initially assigned an ``opinion,'' either 0 or 1. At exponential times, vertices update their values by assuming the majority value of their neighbors. This model has been studied extensively on Z^d, where it is known as the zero-temperature limit of Ising Glauber dynamics. I will review some of the major questions and conjectures on lattices, and then explain some new work with Arnab Sen (Minnesota) on the 3-regular tree. We relate the majority vote model to a new model, which we call the median process, and use this process to answer questions about the limiting state of opinions. For example, we show that when the initial state is given by a Bernoulli(p) product measure, the probability that a vertex's limiting opinion is 1 is a continuous function of p.

The circular law for very sparse random matrices

Series
Stochastics Seminar
Time
Thursday, September 13, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Konstantin Tikhomirov – School of Mathematics, GaTech
Let (A_n) be a sequence of random matrices, such that for every n, A_n is n by n with i.i.d. entries, and each entry is of the form b*x, where b is a Bernoulli random variable with probability of success p_n, and x is an independent random variable of unit variance. We show that, as long as n*p_n converges to infinity, the appropriately rescaled spectral distribution of A_n converges to the uniform measure on the unit disc of complex plane. Based on joint work with Mark Rudelson.

Variational Analysis of Empirical Risk Minimization

Series
Stochastics Seminar
Time
Thursday, August 30, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew Nobel – University of North Carolina, Chapel Hill
This talk concerns the description and analysis of a variational framework for empirical risk minimization. In its most general form the framework concerns a two-stage estimation procedure in which (i) the trajectory of an observed (but unknown) dynamical system is fit to a trajectory from a known reference dynamical system by minimizing average per-state loss, and (ii) a parameter estimate is obtained from the initial state of the best fit reference trajectory. I will show that the empirical risk of the best fit trajectory converges almost surely to a constant that can be expressed in variational form as the minimal expected loss over dynamically invariant couplings (joinings) of the observed and reference systems. Moreover, the family of joinings minimizing the expected loss fully characterizes the asymptotic behavior of the estimated parameters. I will illustrate the breadth of the variational framework through applications to the well-studied problems of maximum likelihood estimation and non-linear regression, as well as the analysis of system identification from quantized trajectories subject to noise, a problem in which the models themselves exhibit dynamical behavior across time.

Asymptotics in random balls models

Series
Stochastics Seminar
Time
Tuesday, June 12, 2018 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jean-Christophe Breton – University of Rennes
Random balls models are collections of Euclidean balls whose centers and radii are generated by a Poisson point process. Such collections model various contexts ranging from imaging to communication network. When the distributions driving the centers and the radii are heavy-tailed, interesting interference phenomena occurs when the model is properly zoomed-out. The talk aims to illustrate such phenomena and to give an overview of the asymptotic behavior of functionals of interest. The limits obtained include in particular stable fields, (fractional) Gaussian fields and Poissonian bridges. Related questions will also be discussed.

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