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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.

Series: Stochastics Seminar

A highlight in the study of quantum physics was the work of Knizhnik, Polyakov and Zamolodchikov (1988), in which they proposed a relation (KPZ relation) between the scaling dimension of a statistical physics model in Euclidean geometry and its counterpart in the random geometry. More recently, Duplantier and Sheffield (2011) used the 2-dim Gaussian free field to construct the Liouville quantum gravity measure on a planar domain, and gave the first mathematically rigorous formulation and proof of the KPZ relation in that setting. Inspired by the work of Duplantier and Sheffield, we apply a similar approach to extend their results and techniques to higher even dimensions R^(2n) for n>=2. This talk mainly focuses on the case of R^4. I will briefly introduce the notion of Gaussian free field (GFF). In our work we adopt a specific 4-dim GFF to construct a random Borel measure on R^4 which formally has the density (with respect to the Lebesgue measure) being the exponential of an instance of the GFF. Further we establish a 4-dim KPZ relation corresponding to this random measure. This work is joint with Dmitry Jakobson (McGill University).

Series: Stochastics Seminar

The limiting spectral distributions of many sparse random graph models are known to contain atoms. But do they also have some continuous part? In this talk, I will give affirmative answer to this question for several widely studied models of random graphs including Erdos-Renyi random graph G(n,c/n) with c > 1, random graphs with certain degree distributions and supercritical bond percolation on Z^2. I will also present several open problems. This is joint work with Charles Bordenave and Balint Virag.

Series: Stochastics Seminar

The 2-core of a hypergraph is the unique subgraph where all vertices have
degree at least 2 and which is the maximal induced subgraph with this
property. This talk will be about the investigation of the 2-core for a
particular random hypergraph model --- a model which differs from the usual
random uniform hypergraph in that the vertex degrees are not identically
distributed.
For this model the main result proved is that as the size of the vertex
set, n, tends to infinity then the number of hyperedges in the 2-core obeys
a limit law, and this limit exhibits a threshold where the number of
hyperedges in the 2-core transitions from o(n) to Theta(n). We will
discuss aspects of the ideas involved and discuss the background motivation
for the hypergraph model: factoring random integers into primes.