- You are here:
- GT Home
- Home
- News & Events

Series: Stochastics Seminar

Burgers turbulence is the study of Burgers equation with random initial data or forcing. While having its origins in hydrodynamics, this model has remarkable connections to a variety of seemingly unrelated problems in statistics, kinetic theory, random matrices, and integrable systems. In this talk I will survey these connections and discuss the crucial role that exact solutions have played in the development of the theory.

Series: Stochastics Seminar

The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers,particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.

Series: Stochastics Seminar

Rank reduction as an effective technique for dimension reduction is
widely used in statistical modeling and machine learning. Modern
statistical applications entail high dimensional data analysis where
there may exist a large number of nuisance variables. But the plain rank
reduction cannot discern relevant or important variables. The talk
discusses joint variable and rank selection for predictive learning. We
propose to apply sparsity and reduced rank techniques to attain
simultaneous feature selection and feature extraction in a vector
regression setup. A class of estimators is introduced based on novel
penalties that impose both row and rank restrictions on the coefficient
matrix. Selectable principle component analysis is proposed and studied
from a self-regression standpoint which gives an extension to the sparse
principle component analysis. We show that these estimators adapt to the
unknown matrix sparsity and have fast rates of convergence in comparison
with LASSO and reduced rank regression. Efficient computational
algorithms are developed and applied to real world applications.

Series: Stochastics Seminar

Cramér's theorem from 1936 states that the sum of two independent random variables is Gaussian if and only if these random variables are Gaussian. Since then, this property has been explored in different directions, such as for other distributions or non-commutative random variables. In this talk, we will investigate recent results in Gaussian chaoses and free chaoses. In particular, we will give a first positive Cramér type result in a free probability context.

Series: Stochastics Seminar

Ricci flow is a sort of (nonlinear) heat problem under which the metric on a given manifold is evolving. There is a deep connection between probability and heat equation. We try to setup a probabilistic approach in the framework of a stochastic target problem. A major result in the Ricci flow is that the normalized flow (the one in which the area is preserved) exists for all positive times and it converges to a metric of constant curvature. We reprove this convergence result in the case of surfaces of non-positive Euler characteristic using coupling ideas from probability. At certain point we need to estimate the second derivative of the Ricci flow and for that we introduce a coupling of three particles. This is joint work with Rob Neel.

Series: Stochastics Seminar

In the context of a linear model with a sparse coefficient vector, sharp oracle inequalities have been established for the exponential weights concerning the prediction problem. We show that such methods also succeed at variable selection and estimation under near minimum condition on the design matrix, instead of much stronger assumptions required by other methods such as the Lasso or the Dantzig Selector. The same analysis yields consistency results for Bayesian methods and BIC-type variable selection under similar conditions. Joint work with Ery Arias-Castro

Series: Stochastics Seminar

The Burgers equation is a basic hydrodynamic model describing the evolution of the velocity field of sticky dust particles. When supplied with random forcing it turns into an infinite-dimensional random dynamical system that has been studied since late 1990's. The variational approach to Burgers equation allows to study the system by analyzing optimal paths in the random landscape generated by the random force potential. Therefore, this is essentially a random media problem. For a long time only compact cases of Burgers dynamics on the circle or a torus were understood well. In this talk I discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption. The main result is the description of the ergodic components and One Force One Solution principle on each component. Joint work with Eric Cator and Kostya Khanin.

Series: Stochastics Seminar

Singular value decomposition is a widely used tool for dimension
reduction in multivariate analysis. However, when used for statistical
estimation in high-dimensional low rank matrix models, singular vectors of
the noise-corrupted matrix are inconsistent for their counterparts of the
true mean matrix. In this talk, we suppose the true singular vectors have
sparse representations in a certain basis. We propose an iterative
thresholding algorithm that can estimate the subspaces spanned by leading
left and right singular vectors and also the true mean matrix optimally
under Gaussian assumption. We further turn the algorithm into a practical
methodology that is fast, data-driven and robust to heavy-tailed noises.
Simulations and a real data example further show its competitive
performance. This is a joint work with Andreas Buja and Dan Yang.

Series: Stochastics Seminar

Many of the asymptotic spectral characteristics of a symmetric random
matrix with i.i.d. entries (such a matrix is called a "Wigner matrix") are
said to be "universal": they depend on the exact distribution of the
entries only via its first moments (in the same way that the CLT gives the
asymptotic fluctuations of the empirical mean of i.i.d. variables as a
function of their second moment only). For example, the empirical spectral
law of the eigenvalues of a Wigner matrix converges to the semi-circle law
if the entries have variance 1, and the extreme eigenvalues converge to -2
and 2 if the entries have a finite fourth moment. This talk will be devoted
to a "universality result" for the eigenvectors of such a matrix. We shall
prove that the asymptotic global fluctuations of these eigenvectors depend
essentially on the moments with orders 1, 2 and 4 of the entries of the
Wigner matrix, the third moment having surprisingly no influence.

Series: Stochastics Seminar

This is obtained as a limit from the classical Poincar\'e on large random matrices. In the classical case Poincare is obtained in a rather easy way from other functional inequalities as for instance Log-Sobolev and transportation. In the free case, the same story becomes more intricate. This is joint work with Michel Ledoux.