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Series: Stochastics Seminar

We consider stochastic dynamical systems defined on networks that exhibit the phenomenon of collective metastability---by this we mean network dynamics where none of the individual nodes' dynamics are metastable, but the configuration is metastable in its collective behavior. We will concentrate on the case of SDE with small white noise for concreteness. We also present some specific results relating to stochastic perturbations of the Kuramoto system of coupled nonlinear oscillators. Along the way, we show that there is a non-standard spectral problem that appears in all of these calculations, and that the important features of this spectral problem is related to a certain homology group.

Series: Stochastics Seminar

I will talk about two model problem concerning a diffusion with a cellular drift (a.k.a array of opposing vortices). The first concerns the expected exit time from a domain as both the flow amplitude $A$ (or more precisely the Peclet number) goes to infinity, AND the cell size (or vortex seperation) $\epsilon$ approaches $0$ simultaneously. When one of the parameters is fixed, the problem has been extensively studied and the limiting behaviour is that of an effective "homogenized" or "averaged" problem. When both vary simultaneously one sees an interesting transition at $A \approx \eps^{-4}$. While the behaviour in the averaged regime ($A \gg \eps^{-4}$) is well understood, the behaviour in the homogenized regime ($A \ll \eps^{-4}$) is poorly understood, and the critical transition regime is not understood at all. The second problem concerns an anomalous diffusive behaviour observed in "intermediate" time scales. It is well known that a passive tracer diffusing in the presence of a strong cellular flows "homogenizes" and behaves like an effective Brownian motion on large time scales. On intermediate time scales, however, an anomalous diffusive behaviour was numerically observed recently. I will show a few preliminary rigorous results indicating that the stable "anomalous" behaviour at intermediate time scales is better modelled through Levy flights, and show how this can be used to recover the homogenized Brownian behaviour on long time scales.

Series: Stochastics Seminar

I will review recent progress concerning nonparametric estimation of log-concave densities and related families in $R^1$ and $R^d$. In the case of $R^1$, I will present limit theory for the estimators at fixed points at which the population density has a non-zero second derivative and for the resulting natural mode estimator under a corresponding hypothesis. In the case of $R^d$ with $d\ge 2$ will briefly discuss some recent progress and sketch a variety of open problems.

Series: Stochastics Seminar

In this talk I will describe a theory of matrix completion for the extreme
case of noisy 1-bit observations. In this setting, instead of observing a
subset of the real-valued entries of a matrix M, we obtain a small number
of binary (1-bit) measurements generated according to a probability
distribution determined by the real-valued entries of M. The central
question I will address is whether or not it is possible to obtain an
accurate estimate of M from this data. In general this would seem
impossible, but I will show that the maximum likelihood estimate under a
suitable constraint returns an accurate estimate of M when $\|M\|_{\infty}
\le \alpha$ and $\rank(M) \le r$. If the log-likelihood is a concave
function (e.g., the logistic or probit observation models), then we can
obtain this maximum likelihood estimate by optimizing a convex program. I
will also provide lower bounds showing that this estimate is near-optimal
and illustrate the potential of this method with some preliminary numerical
simulations.

Series: Stochastics Seminar

The periodic generalized Korteweg-de Vries equation (gKdV) is a canonical dispersive partial differential equation with numerous applications in physics and engineering. In this talk we present invariance of the Gibbs measure under the flow of the gauge transformed periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized. As a corollary we obtain almost sure global well-posedness for the (ungauged) quartic gKdV at regularities where this PDE is deterministically ill-posed.

Series: Stochastics Seminar

In small dimension a random geometric graph behaves
very differently from a standard Erdös-Rényi random graph. On the other
hand when the dimension tends to infinity (with the number of vertices being
fixed) both models coincides. In this talk we study the behavior of the clique
number of random geometric graphs when the dimension grows with the
number of vertices.

Series: Stochastics Seminar

Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the 2D Gaussian free field, are conjectured to form a new universality class of extreme value statistics (notably in the work of Carpentier & Ledoussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will report on the recent rigorous progress in describing the new features of this class. In particular, I will describe the emergence of Poisson-Dirichlet statistics. This is joint work with Olivier Zindy.

Series: Stochastics Seminar

There is a long history on the study of zeros of random polynomials whose coefficients are independent, identically distributed, non-degenerate random variables. We will first provide an overview on zeros of random functions and then show exact and/or asymptotic bounds on probabilities that all zeros of a random polynomial are real under various distributions. The talk is accessible to undergraduate and graduate students in any areas of mathematics.

Series: Stochastics Seminar

Smoothness is a fundamental principle in the study of measures on infinite-dimensional spaces, where an obvious obstruction to overcome is the lack of an infinite-dimensional Lebesgue or volume measure. Canonical examples of smooth measures include those induced by a Brownian motion, both its end point distribution and as a real-valued path. More generally, any Gaussian measure on a Banach space is smooth. Heat kernel measure is the law of a Brownian motion on a curved space, and as such is the natural analogue of Gaussian measure there. We will discuss some recent smoothness results for these measures on certain classes of infinite-dimensional groups, including in some degenerate settings. This is joint work with Fabrice Baudoin, Daniel Dobbs, and Masha Gordina.

Series: Stochastics Seminar

This paper concerns the problem of matrix completion, which is to
estimate a matrix from observations in a small subset of indices. We
propose a calibrated spectrum elastic net method with a sum of the
nuclear and Frobenius penalties and develop an iterative algorithm to
solve the convex minimization problem. The iterative algorithm
alternates between imputing the missing entries in the incomplete matrix
by the current guess and estimating the matrix by a scaled
soft-thresholding singular value decomposition of the imputed matrix
until the resulting matrix converges. A calibration step follows to
correct the bias caused by the Frobenius penalty. Under proper coherence
conditions and for suitable penalties levels, we prove that the proposed estimator achieves an error bound of nearly optimal order and in proportion to the noise level. This provides a unified analysis of the noisy and noiseless matrix completion problems.
Tingni Sun and Cun-Hui Zhang, Rutgers University