Cancelled
- Series
- Stochastics Seminar
- Time
- Thursday, March 6, 2014 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Ioana Dumirtiu – Univ. of Washington
- You are here:
- Home
- News & Events
Asymptotics of spectral projectors of sample covariance
- Series
- Stochastics Seminar
- Time
- Thursday, February 27, 2014 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Vladimir Koltchinskii – Gatech
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.
Information Relaxation and Duality in Stochastic Optimal Control
- Series
- Stochastics Seminar
- Time
- Thursday, February 6, 2014 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Enlu Zhou – ISYE Gatech
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.
Random matrix theory and the informational limit of eigen-analysis
- Series
- Stochastics Seminar
- Time
- Thursday, December 12, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Raj Rao Nadakuditi – University of Michigan
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.
Distributions of Angles in Random Packing on Spheres
- Series
- Stochastics Seminar
- Time
- Thursday, December 5, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Tiefeng Jiang – University of Minnesota
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.
Gaussian free field, random measure and KPZ on R^4
- Series
- Stochastics Seminar
- Time
- Thursday, November 21, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Linan Chen – McGill University
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).
Continuous spectra for sparse random graphs
- Series
- Stochastics Seminar
- Time
- Thursday, November 14, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Arnab Sen – University of Minnesota
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.
The 2-core of a Random Inhomogeneous Hypergraph
- Series
- Stochastics Seminar
- Time
- Thursday, November 7, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Omar Abuzzahab – Georgia Tech
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.
Thresholds for Random Geometric k-SAT
- Series
- Stochastics Seminar
- Time
- Thursday, October 24, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Will Perkins – Georgia Tech, School of Mathematics
Random k-SAT is a distribution over boolean formulas studied widely in both statistical physics and theoretical computer science for its intriguing behavior at its phase transition. I will present results on the satisfiability threshold in a geometric model of random k-SAT: labeled boolean literals are placed uniformly at random in a d-dimensional cube, and for each set of k contained in a ball of radius r, a k-clause is added to the random formula. Unlike standard random k-SAT, this model exhibits dependence between the clauses. For all k we show that the satisfiability threshold is sharp, and for k=2 we find the location of the threshold as well. I will also discuss connections between this model, the random geometric graph, and other probabilistic models. This is based on joint work with Milan Bradonjic.
Large Average Submatrices of a Gaussian Random Matrix: Landscapes and Local Optima
- Series
- Stochastics Seminar
- Time
- Thursday, October 10, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skyles 005
- Speaker
- Andrew Nobel – University of North Carolina, Chapel Hill
The problem of finding large average submatrices of a real-valued matrix arises in the exploratory analysis of data from disciplines as diverse as genomics and social sciences. Motivated in part by previous work on this applied problem, this talk will present several new theoretical results concerning large average submatrices of an n x n Gaussian random matrix. We will begin by considering the average and joint distribution of the k x k submatrix having largest average value (the global maximum). We then turn our attention to submatrices with dominant row and column sums, which arise as the local maxima of a practical iterative search procedure for large average submatrices I will present a result characterizing the value and joint distribution of a local maximum, and show that a typical local maxima has an average value within a constant factor of the global maximum. In the last part of the talk I will describe several results concerning the *number* L_n(k) of k x k local maxima, including the asymptotic behavior of its mean and variance for fixed k and increasing n, and a central limit theorem for L_n(k) that is based on Stein's method for normal approximation.
Joint work with Shankar Bhamidi (UNC) and Partha S. Dey (UIUC)