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Series: Stochastics Seminar

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".

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

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.

Series: Stochastics Seminar

Series: Stochastics Seminar

Several new results on asymptotic normality
and other asymptotic properties of sample covariance operators
for Gaussian observations in a high-dimensional
setting will be discussed. Such asymptotics are of importance
in various problems of high-dimensional statistics (in particular,
related to principal component analysis). The proofs of these results
rely on Gaussian concentration inequality. This is a joint work
with Karim Lounici.

Series: Stochastics Seminar

In this talk, I will talk about some recent research development in the approach of information relaxation to explore duality in Markov decision processes and controlled Markov diffusions. The main idea of information relaxation is to relax the constraint that the decisions should be made based on the current information and impose a penalty to punish the access to the information in advance. The weak duality, strong duality and complementary slackness results are then established, and the structures of optimal penalties are revealed. The dual formulation is essentially a sample path-wise optimization problem, which is amenable to Monte Carlo simulation. The duality gap associated with a sub-optimal policy/solution also gives a practical indication of the quality of the policy/solution.

Series: Stochastics Seminar

Motivated by the ubiquity of signal-plus-noise type models in high-dimensional statistical signal processing and machine learning, we consider the eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Applications in mind are as diverse as radar, sonar, wireless communications, spectral clustering, bio-informatics and Gaussian mixture cluster analysis in machine learning. We provide an application-independent approach that brings into sharp focus a fundamental informational limit of high-dimensional eigen-analysis. Building on this success, we highlight the random matrix origin of this informational limit, the connection with "free" harmonic analysis and discuss how to exploit these insights to improve low-rank signal matrix denoising relative to the truncated SVD.

Series: Stochastics Seminar

We study the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in p-dimensional spaces as the number of points n goes to infinity, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that ``all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and connections with some open problems in physics and mathematics are also discussed. This is a joint work with Tony Cai and Jianqing Fan.