Thursday, March 27, 2014 - 15:05 , Location: Skiles 005 , Robert Neel , Lehigh Univ. , Organizer: Ionel Popescu
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.
Thursday, March 13, 2014 - 15:05 , Location: Skiles 005 , Konstantinos Spiliopoulos , Boston University , Organizer:
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.
Thursday, February 27, 2014 - 15:05 , Location: Skiles 005 , Vladimir Koltchinskii , Gatech , Organizer: Ionel Popescu
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.
Thursday, February 6, 2014 - 15:05 , Location: Skiles 005 , Enlu Zhou , ISYE Gatech , Organizer: Ionel Popescu
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.
Thursday, December 12, 2013 - 15:05 , Location: Skiles 005 , Raj Rao Nadakuditi , University of Michigan , Organizer: Ionel Popescu
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.
Thursday, December 5, 2013 - 15:05 , Location: Skiles 005 , Tiefeng Jiang , University of Minnesota , Organizer: Ionel Popescu
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.
Thursday, November 21, 2013 - 15:05 , Location: Skiles 005 , Linan Chen , McGill University , Organizer: Ionel Popescu
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).
Thursday, November 14, 2013 - 15:05 , Location: Skiles 005 , Arnab Sen , University of Minnesota , Organizer: Ionel Popescu
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.
Thursday, November 7, 2013 - 15:05 , Location: Skiles 005 , Omar Abuzzahab , Georgia Tech , Organizer: Ionel Popescu
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.