Thursday, September 6, 2012 - 15:05 , Location: Skiles 006 , Yuri Bakhtin , Georgia Tech , Organizer:
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.
Tuesday, April 24, 2012 - 16:05 , Location: skyles 006 , Zongming Ma , The Wharton School, Department of Statistics, University of Pennsylvania , Organizer: Karim Lounici
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.
Tuesday, April 24, 2012 - 11:00 , Location: Skiles 005 , F. Benaych-Georges , Universite Pierre et Marie Curie , Organizer: Christian Houdre
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.
Thursday, April 19, 2012 - 15:05 , Location: Skyles 006 , Ionel Popescu , Georgia Institute of Technology, School of Mathematics , Organizer: Karim Lounici
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.
Tuesday, April 3, 2012 - 16:05 , Location: Skyles 006 , Axel Munk , Institut für Mathematische Stochastik Georg-August-Universität Göttingen , Organizer: Karim Lounici
In this talk we will discuss a general concept of statistical multiscale analysis in the context of signal detection and imaging. This provides a large class of fully data driven regularisation methods which can be viewed as a multiscale generalization of the Dantzig selector. We address computational issues as well as the required extreme value theory of the multiscale statistics. Two major example include change point regression and locally adaptive total variation image regularization for deconvolution problems. Our method is applied to problems from ion channel recordings and nanoscale biophotonic cell microscopy.
Thursday, March 29, 2012 - 15:05 , Location: skyles 006 , Alexander Rakhlin , University of Pennsylvania, The Wharton School , Organizer: Karim Lounici
The study of prediction within the realm of Statistical Learning Theory is intertwined with the study of the supremum of an empirical process. The supremum can be analyzed with classical tools:Vapnik-Chervonenkis and scale-sensitive combinatorial dimensions, covering and packing numbers, and Rademacher averages. Consistency of empirical risk minimization is known to be closely related to theuniform Law of Large Numbers for function classes.In contrast to the i.i.d. scenario, in the sequential prediction framework we are faced with an individual sequence of data on which weplace no probabilistic assumptions. The problem of universal prediction of such deterministic sequences has been studied withinStatistics, Information Theory, Game Theory, and Computer Science. However, general tools for analysis have been lacking, and mostresults have been obtained on a case-by-case basis.In this talk, we show that the study of sequential prediction is closely related to the study of the supremum of a certain dyadic martingale process on trees. We develop analogues of the Rademacher complexity, covering numbers and scale-sensitive dimensions, which canbe seen as temporal generalizations of the classical results. The complexities we define also ensure uniform convergence for non-i.i.d. data, extending the Glivenko-Cantelli type results. Analogues of local Rademacher complexities can be employed for obtaining fast rates anddeveloping adaptive procedures. Our understanding of the inherent complexity of sequential prediction is complemented by a recipe that can be used for developing new algorithms.
Thursday, March 15, 2012 - 15:05 , Location: skyles 006 , Vygantas Paulauskas , Vilnius University, Lithuania , Organizer: Karim Lounici
In the talk we demonstrate the usefulness of the so-called Beveridge-Nelson decomposition in asymptotic analysis of sums of values of linear processes and fields. We consider several generalizations of this decomposition and discuss advantages and shortcomings of this approach which can be considered as one of possible methods to deal with sums of dependent random variables. This decomposition is derived for linear processes and fields with the continuous time (space) argument. The talk is based on several papers, among them [V. Paulauskas, J. Multivar. Anal. 101, (2010), 621-639] and [Yu. Davydov and V. Paulauskas, Teor. Verojat. Primenen., (2012), to appear]
Thursday, March 1, 2012 - 15:05 , Location: Skyles 006 , Gregorio Moreno Flores , University of Wisconsin, department of Mathematics , Organizer: Karim Lounici
The usual approach to KPZ is to study the scaling limit of particle systems. In this work, we show that the partition function of directed polymers (with a suitable boundary condition) converges, in a certain regime, to the Cole-Hopf solution of the KPZ equation in equilibrium. Coupled with some bounds on the fluctuations of directed polymers, this approach allows us to recover the cube root fluctuation bounds for KPZ in equilibrium. We also discuss some partial results for more general initial conditions.
Thursday, February 23, 2012 - 15:05 , Location: 006 , Robert W. Neel , Lehigh University , email@example.com , Organizer: Ionel Popescu
We wish to understand ends of minimal surfaces contained in certain subsets of R^3. In particular, after explaining how the parabolicity and area growth of such minimal ends have been previously studied using universal superharmonic functions, we describe an alternative approach, yielding stronger results, based on studying Brownian motion on the surface. It turns out that the basic results also apply to a larger class of martingales than Brownian motion on a minimal surface, which both sheds light on the underlying geometry and potentially allows applications to other problems.
Thursday, February 16, 2012 - 15:05 , Location: Skyles 006 , Oren Louidor , UCLA , Organizer: Karim Lounici
We consider random walks on Z^d among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but which can be arbitrarily close to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point after time 2n. We show that in the situations when the heat kernel exhibits subdiffusive behavior --- which is known to be possible in dimensions d \geq 4-- the walk gets trapped for time of order n in a small spatial region. This proves that the strategy used to infer subdiffusive lower bounds on the heat kernel in earlier studies of this problem is in fact dominant. In addition, we settle a conjecture on the maximal possible subdiffusive decay in four dimensions and prove that anomalous decay is a tail and thus zero-one event. Joint work with Marek Biskup, Alexander Vandenberg and Alexander Rozinov.