## Seminars and Colloquia by Series

### Learning Gaussian mixtures with algebraic structure

Series
Stochastics Seminar
Time
Thursday, April 22, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/129119189
Speaker
Victor-Emmanuel BrunelENSAE/CREST

We will consider a model of mixtures of Gaussian distributions, called Multi-Reference Alignment, which has been motivated by imaging techniques in chemistry. In that model, the centers are all related with each other by the action of a (known) group of isometries. In other words, each observation is a noisy version of an isometric transformation of some fixed vector, where the isometric transformation is taken at random from some group of isometries and is not observed. Our goal is to learn that fixed vector, whose orbit by the action of the group determines the set of centers of the mixture. First, we will discuss the asymptotic performances of the maximum-likelihood estimator, exhibiting two scenarios that yield different rates. We will then move on to a non-asymptotic, minimax approach of the problem.

### The parking model in Z^d

Series
Stochastics Seminar
Time
Thursday, April 15, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
David SivakoffThe Ohio State University

At each site of Z^d, initially there is a car with probability p or a vacant parking spot with probability (1-p), and the choice is independent for all sites. Cars perform independent simple, symmetric random walks, which do not interact directly with one another, and parking spots do not move. When a car enters a site that contains a vacant spot, then the car parks at the spot and the spot is filled – both the car and the spot are removed from the system, and other cars can move freely through the site. This model exhibits a phase transition at p=1/2: all cars park almost surely if and only if p\le 1/2, and all vacant spots are filled almost surely if and only if p \ge 1/2. We study the rates of decay of cars and vacant spots at, below and above p=1/2. In many cases these rates agree with earlier findings of Bramson—Lebowitz for two-type annihilating systems wherein both particle types perform random walks at equal speeds, though we identify significantly different behavior when p<1/2. Based on joint works with Damron, Gravner, Johnson, Junge and Lyu.

Online at https://bluejeans.com/129119189

### On a conjectural symmetric version of the Ehrhard inequality

Series
Stochastics Seminar
Time
Thursday, April 8, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Galyna LivshytsGeorgiaTech

We will discuss a conjectured sharp version of an Ehrhard-type inequality for symmetric convex sets, its connections to other questions, and partial progress towards it. We also discuss some new estimates for non-gaussian measures.

### Optimal Ranking Recovery from Pairwise Comparisons

Series
Stochastics Seminar
Time
Thursday, April 1, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/129119189
Speaker
Anderson Y. ZhangUniversity of Pennsylvania

Ranking from pairwise comparisons is a central problem in a wide range of learning and social contexts. Researchers in various disciplines have made significant methodological and theoretical contributions to it. However, many fundamental statistical properties remain unclear especially for the recovery of ranking structure. This talk presents two recent projects towards optimal ranking recovery, under the Bradley-Terry-Luce (BTL) model.

In the first project, we study the problem of top-k ranking. That is, to optimally identify the set of top-k players. We derive the minimax rate and show that it can be achieved by MLE. On the other hand, we show another popular algorithm, the spectral method, is in general suboptimal.

In the second project, we study the problem of full ranking among all players. The minimax rate exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. A divide-and-conquer ranking algorithm is proposed to achieve the minimax rate.

### TBA by Matthias Löffler

Series
Stochastics Seminar
Time
Thursday, March 11, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Online TBA
Speaker
Matthias LöfflerETH Zurich

### Large Values of the Riemann Zeta Function in Small Intervals

Series
Stochastics Seminar
Time
Thursday, February 25, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Louis-Pierre ArguinBaruch College, CUNY

I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function in small intervals of the critical line. This problem has interesting connections with the extreme value statistics of IID and log-correlated random variables.

### Lower bounds for the estimation of principal components

Series
Stochastics Seminar
Time
Thursday, February 4, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Martin WahlHumboldt University in Berlin

This talk will be concerned with nonasymptotic lower bounds for the estimation of principal subspaces. I will start by reviewing some previous methods, including the local asymptotic minimax theorem and the Grassmann approach. Then I will present a new approach based on a van Trees inequality (i.e. a Bayesian version of the Cramér-Rao inequality) tailored for invariant statistical models. As applications, I will provide nonasymptotic lower bounds for principal component analysis and the matrix denoising problem, two examples that are invariant with respect to the orthogonal group. These lower bounds are characterized by doubly substochastic matrices whose entries are bounded by the different Fisher information directions, confirming recent upper bounds in the context of the empirical covariance operator.

### The Bulk and the Extremes of Minimal Spanning Acycles and Persistence Diagrams of Random Complexes

Series
Stochastics Seminar
Time
Thursday, January 21, 2021 - 15:30 for 1 hour (actually 50 minutes)
Location
Speaker
Sayan MukherjeeDuke University

Frieze showed that the expected weight of the minimum spanning tree (MST) of the uniformly weighted graph converges to ζ(3). Recently, this result was extended to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analogue -- the Minimum Spanning Acycle (MSA). In this work, we go beyond and look at the histogram of the weights in this random MSA -- both in the bulk and in the extremes. In particular, we focus on the `incomplete' setting, where one has access only to a fraction of the potential face weights. Our first result is that the empirical distribution of the MSA weights asymptotically converges to a measure based on the shadow -- the complement of graph components in higher dimensions. As far as we know, this result is the first to explore the connection between the MSA weights and the shadow. Our second result is that the extremal weights converge to an inhomogeneous Poisson point process. A interesting consequence of our two results is that we can also state the distribution of the death times in the persistence diagram corresponding to the above weighted complex, a result of interest in applied topology.

Based on joint work with Nicolas Fraiman and Gugan Thoppe, see https://arxiv.org/abs/2012.14122

### A Lévy-driven process with matrix scaling exponent

Series
Stochastics Seminar
Time
Thursday, December 3, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/504188361
Speaker
B. Cooper BonieceWashington University in St. Louis

In the past several decades, scale invariant stochastic processes have been used in a wide range of applications including internet traffic modeling and hydrology.  However, by comparison to univariate scale invariance, far less attention has been paid to characteristically multivariate models that display aspects of scaling behavior the limit theory arguably suggests is most natural.

In this talk, I will introduce a new scale invariance model called operator fractional Lévy motion and discuss some of its interesting features, as well as some aspects of wavelet-based estimation of its scaling exponents. This is related to joint work with Gustavo Didier (Tulane University), Herwig Wendt (CNRS, IRIT Univ. of Toulouse) and Patrice Abry (CNRS, ENS-Lyon).

### New Classes of Multivariate Covariance Functions

Series
Stochastics Seminar
Time
Thursday, November 19, 2020 - 15:30 for 1 hour (actually 50 minutes)
Location
https://gatech.webex.com/gatech/j.php?MTID=mee147c52d7a4c0a5172f60998fee267a
Speaker
Tatiyana ApanasovichGeorge Washington University

The class which is refereed to as the Cauchy family allows for the simultaneous modeling of the long memory dependence and correlation at short and intermediate lags. We introduce a valid parametric family of cross-covariance functions for multivariate spatial random fields where each component has a covariance function from a Cauchy family. We present the conditions on the parameter space that result in valid models with varying degrees of complexity. Practical implementations, including reparameterizations to reflect the conditions on the parameter space will be discussed. We show results of various Monte Carlo simulation experiments to explore the performances of our approach in terms of estimation and cokriging. The application of the proposed multivariate Cauchy model is illustrated on a dataset from the field of Satellite Oceanography.

Link to Cisco Webex meeting: https://gatech.webex.com/gatech/j.php?MTID=mee147c52d7a4c0a5172f60998fee267a