Seminars and Colloquia by Series

Ultra sub-Gaussian random vectors and Khinchine type inequalities

Series
Stochastics Seminar
Time
Thursday, February 12, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Piotr Nayar – IMA, Minneapolis –
We define the class of ultra sub-Gaussian random vectors and derive optimal comparison of even moments of linear combinations of such vectors in the case of the Euclidean norm. In particular, we get optimal constants in the classical Khinchine inequality. This is a joint work with Krzysztof Oleszkiewicz.

Estimation of convex bodies

Series
Stochastics Seminar
Time
Friday, October 3, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Victor-Emmanuel Brunel – CREST and Yale University
In this talk we will consider a finite sample of i.i.d. random variables which are uniformly distributed in some convex body in R^d. We will propose several estimators of the support, depending on the information that is available about this set: for instance, it may be a polytope, with known or unknown number of vertices. These estimators will be studied in a minimax setup, and minimax rates of convergence will be given.

Second order free CLT

Series
Stochastics Seminar
Time
Thursday, September 25, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ionel Popescu – Georgia Tech
The CLT for free random variables was settled by Voiculescu very early in this work on free probability. He used this in turn to prove his main result on aymptotic freeness of independent random matrices. On the other hand, in random matrices, fluctuations can be understood as a second order phenomena. This notion of fluctuations has a conterpart in free probability which is called freenes of second order. I will explain what this is and how one can prove a free CLT result in this context. It is also interesting to point out that this is a nontrivial calculation which begs the same question in the classical context and I will comment on that.

Moment bounds and concentration for sample covariance operators in Banach spaces

Series
Stochastics Seminar
Time
Thursday, September 11, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Vladimir Koltchinskii – School of Mathematics, Georgia Tech
We will discuss sharp bounds on moments and concentration inequalities for the operator norm of deviations of sample covariance operators from the true covariance operator for i.i.d. Gaussian random variables in a separable Banach space. Based on a joint work with Karim Lounici.

A Central Limit Theorem for the Length of the Longest Common Subsequence in Random Words

Series
Stochastics Seminar
Time
Thursday, September 4, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christian Houdre – School of Mathematics, Georgia Tech
Let (X_k)_{k \geq 1} and (Y_k)_{k\geq1} be two independent sequences of independent identically distributed random variables having the same law and taking their values in a finite alphabet \mathcal{A}_m. Let LC_n be the length of the longest common subsequence of the random words X_1\cdots X_n and Y_1\cdots Y_n. Under assumptions on the distribution of X_1, LC_n is shown to satisfy a central limit theorem. This is in contrast to the Bernoulli matching problem or to the random permutations case, where the limiting law is the Tracy-Widom one. (Joint with Umit Islak)

Sandpiles and system-spanning avalanches

Series
Stochastics Seminar
Time
Thursday, April 17, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lionel Levine – Cornell University
A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s_0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a sandpile s_\tau that topples forever. Statistical physicists Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in this "threshold state" s_\tau in the limit as s_0 goes to negative infinity. I will outline the proof of this conjecture in http://arxiv.org/abs/1402.3283 and explain the big-picture motivation, which is to give more predictive power to the theory of "self-organized criticality".

The Gaussian Radon transform for Banach spaces and machine learning

Series
Stochastics Seminar
Time
Thursday, April 10, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Irina Holmes – Louisiana State University
In this talk we investigate possible applications of the infinitedimensional Gaussian Radon transform for Banach spaces to machine learning. Specifically, we show that the Gaussian Radon transform offers a valid stochastic interpretation to the ridge regression problem in the case when the reproducing kernel Hilbert space in question is infinite-dimensional. The main idea is to work with stochastic processes defined not on the Hilbert space itself, but on the abstract Wiener space obtained by completing the Hilbert space with respect to a measurable norm.

Concentration Inequalities with Bounded Couplings

Series
Stochastics Seminar
Time
Thursday, April 3, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Umit Islak – University of Southern California
Let $Y$ be a nonnegative random variable with mean $\mu$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by $\mathbb{E}[Yf(Y)]=\mu \mathbb{E}[f(Y^s)]$ for all functions $f$ for which these expectations exist. Under bounded coupling conditions, such as $Y^s-Y \leq C$ for some $C>0$, we show that $Y$ satisfies certain concentration inequalities around $\mu$. Examples will focus on occupancy models with log-concave marginal distributions.

Heat kernel asymptotics at the cut locus

Series
Stochastics Seminar
Time
Thursday, March 27, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert Neel – Lehigh Univ.
We discuss a technique, going back to work of Molchanov, for determining the small-time asymptotics of the heat kernel (equivalently, the large deviations of Brownian motion) at the cut locus of a (sub-) Riemannian manifold (valid away from any abnormal geodesics). We relate the leading term of the expansion to the structure of the cut locus, especially to conjugacy, and explain how this can be used to find general bounds as well as to compute specific examples. We also show how this approach leads to restrictions on the types of singularities of the exponential map that can occur along minimal geodesics. Further, time permitting, we extend this approach to determine the asymptotics for the gradient and Hessian of the logarithm of the heat kernel on a Riemannian manifold, giving a characterization of the cut locus in terms of the behavior of the log-Hessian, which can be interpreted in terms of large deviations of the Brownian bridge. Parts of this work are joint with Davide Barilari, Ugo Boscain, and Grégoire Charlot.

Large deviations and Monte Carlo methods for problems with multiple scales

Series
Stochastics Seminar
Time
Thursday, March 13, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Konstantinos Spiliopoulos – Boston University
Rare events, metastability and Monte Carlo methods for stochastic dynamical systems have been of central scientific interest for many years now. In this talk we focus on multiscale systems that can exhibit metastable behavior, such as rough energy landscapes. We discuss quenched large deviations in related random rough environments and design of provably efficient Monte Carlo methods, such as importance sampling, in order to estimate probabilities of rare events. Depending on the type of interaction of the fast scales with the strength of the noise we get different behavior, both for the large deviations and for the corresponding Monte Carlo methods. Standard Monte Carlo methods perform poorly in these kind of problems in the small noise limit. In the presence of multiple scales one faces additional difficulties and straightforward adaptation of importance sampling schemes for standard small noise diffusions will not produce efficient schemes. We resolve this issue and demonstrate the theoretical results by examples and simulation studies.

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