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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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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]

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.