Seminars and Colloquia by Series

Thursday, February 24, 2011 - 15:05 , Location: Skyles 005 , Ben Recht , Computer Sciences Department, University of Wisconsin , Organizer: Karim Lounici
Deducing the state or structure of a system from partial, noisy measurements is a fundamental task throughout the sciences and engineering. The resulting inverse problems are often ill-posed because there are fewer measurements available than the ambient dimension of the model to be estimated. In practice, however, many interesting signals or models contain few degrees of freedom relative to their ambient dimension: a small number of genes may constitute the signature of a disease, very few parameters may specify the correlation structure of a time series, or a sparse collection of geometric constraints may determine a molecular configuration. Discovering, leveraging, or recognizing such low-dimensional structure plays an important role in making inverse problems well-posed. In this talk, I will propose a unified approach to transform notions of simplicity and latent low-dimensionality into convex penalty functions.  This approach builds on the success of generalizing compressed sensing to matrix completion, and greatly extends the catalog of objects and structures that can be recovered from partial information. I will focus on a suite of data analysis algorithms designed to decompose general signals into sums of atoms from a simple---but not necessarily discrete---set. These algorithms are derived in a convex optimization framework that encompasses previous methods based on l1-norm minimization and nuclear norm minimization for recovering sparse vectors and low-rank matrices. I will provide sharp estimates of the number of generic measurements required for exact and robust recovery of a variety of structured models.  These estimates are based on computing certain Gaussian statistics related to the latent model geometry. I will detail several example applications and describe how to scale the corresponding inference algorithms to very large data sets. (Joint work with Venkat Chandrasekaran, Pablo Parrilo, and Alan Willsky)
Thursday, January 27, 2011 - 16:05 , Location: Skiles 005 , Thomas Lee , University of California, Davis , Organizer: Liang Peng
In this talk we re-visit Fisher's controversial fiducial technique for conducting statistical inference. In particular, a generalization of Fisher's technique, termed generalized fiducial inference, is introduced. We illustrate its use with wavelet regression. Current and future work for generalized fiducial inference will also be discussed. Joint work with Jan Hannig and Hari Iyer
Thursday, January 27, 2011 - 15:05 , Location: Skiles 005 , Ery Arias-Castro , University of California, San Diego , Organizer: Karim Lounici
We study the problem of testing for the significance of a subset of regression coefficients in a linear model under the assumption that the coefficient vector is sparse, a common situation in modern high-dimensional settings.  Assume there are p variables and let S be the number of nonzero coefficients.  Under moderate sparsity levels, when we may have S > p^(1/2), we show that the analysis of variance F-test is essentially optimal.  This is no longer the case under the sparsity constraint S < p^(1/2).  In such settings, a multiple comparison procedure is often preferred and we establish its optimality under the stronger assumption S < p^(1/4).  However, these two very popular methods are suboptimal, and sometimes powerless, when p^(1/4) < S < p^(1/2).  We suggest a method based on the Higher Criticism that is essentially optimal in the whole range S < p^(1/2).  We establish these results under a variety of designs, including the classical (balanced) multi-way designs and more modern `p > n' designs arising in genetics and signal processing. (Joint work with Emmanuel Candès and Yaniv Plan.)
Thursday, December 2, 2010 - 15:05 , Location: Skiles 002 , Santosh Vempala , College of Computing, Georgia Tech , Organizer:
For general graphs, approximating the maximum clique is a notoriously hard problem even to approximate to a factor of nearly n, the number of vertices. Does the situation get better with random graphs? A random graph on n vertices where each edge is chosen with probability 1/2 has a clique of size nearly 2\log n with high probability. However, it is not know how to find one of size 1.01\log n in polynomial time. Does the problem become easier if a larger clique were planted in a random graph? The current best algorithm can find a planted clique of size roughly n^{1/2}. Given that any planted clique of size greater than 2\log n is unique with high probability, there is a large gap here. In an intriguing paper, Frieze and Kannan introduced a tensor-based method that could reduce the size of the planted clique to as small as roughly n^{1/3}. Their method relies on finding the spectral norm of a 3-dimensional tensor, a problem whose complexity is open. Moreover, their combinatorial proof does not seem to extend beyond this threshold. We show how to recover the Frieze-Kannan result using a purely probabilistic argument that generalizes naturally to r-dimensional tensors and allows us recover cliques of size as small as poly(r).n^{1/r} provided we can find the spectral norm of r-dimensional tensors. We highlight the algorithmic question that remains open. This is joint work with Charlie Brubaker.
Thursday, November 18, 2010 - 15:05 , Location: Skiles 002 , Richard Samworth , Statistical Laboratory, Cambridge, UK , Organizer:
If $X_1,...,X_n$ are a random sample from a density $f$ in $\mathbb{R}^d$, then with probability one there exists a unique log-concave maximum likelihood estimator $\hat{f}_n$ of $f$.  The use of this estimator is attractive because, unlike kernel density estimation, the estimator is fully automatic, with no smoothing parameters to choose. We exhibit an iterative algorithm for computing the estimator and show how the method can be combined with the EM algorithm to fit finite mixtures of log-concave densities. Applications to classification, clustering and functional estimation problems will be discussed, as well as recent theoretical results on the performance of the estimator.  The talk will be illustrated with pictures from the R package LogConcDEAD. Co-authors: Yining Chen, Madeleine Cule, Lutz Duembgen (Bern), RobertGramacy (Cambridge), Dominic Schuhmacher (Bern) and Michael Stewart
Thursday, November 11, 2010 - 15:05 , Location: Skiles 002 , Radoslaw Adamczak , University of Warsaw and Fields Institute , Organizer:
I will discuss certain geometric properties of random matrices with independent logarithmically concave columns, obtained in the last several years jointly with O. Guedon, A. Litvak, A. Pajor and N. Tomczak-Jaegermann. In particular I will discuss estimates on the largest and smallest singular values of such matrices and rates on convergence of empirical approximations to covariance matrices of log-concave measures (the Kannan-Lovasz-Simonovits problem).
Thursday, October 21, 2010 - 15:05 , Location: Skiles 002 , Slim Ayadi , School of Math, Georgia Tech , Organizer:
  We study the spectral properties of matrices of long-range percolation model. These are N*N random real symmetric matrices H whose elements are independent random variables taking zero value with probability 1-\psi((i-j)/b), b\in \R^{+}, where \psi is an even positive function with \psi(t)<1 and vanishing at infinity. We show that under rather general conditions on the probability distribution of H(i,j) the semicircle law is valid for the ensemble we study in the limit N,b\to\infty. In the second part, we study the leading term of the correlation function of the resolvent G(z)=(H-z)^{-1} with large enough |Imz| in the limit N,b\to\infty, b=O(N^{\alpha}), 1/3<\alpha<1. We show that this leading term, when considered in the local spectral scale leads to an expression found earlier by other authors for band random matrix ensembles. This shows that the ensemble we study and that of band random matrices belong to the same class of spectral universality. 
Thursday, October 7, 2010 - 15:05 , Location: Skiles 002 , Alexei Novikov , Penn State , Organizer:
The G-equation is a Hamilton-Jacobi level-set equation, that is used in turbulent combustion theory. Level sets of the solution represent a flame surface which moves with normal velocity that is the sum of the laminar flame velocity and the fluid velocity. In this work I will discuss the large-scale long-time asymptotics of these solutions when the fluid velocity is modeled as a stationary incompressible random field. The main challenge of this work comes from the fact that our Hamiltonian is noncoercive. This is a joint work with J.Nolen.
Thursday, September 30, 2010 - 15:05 , Location: Skiles 002 , Amarjit Budhiraja , University of North Carolina at Chapel Hill , Organizer:
A two-player zero-sum stochastic differential game, defined in terms of an m-dimensional state process that is driven by a one-dimensional Brownian motion, played until the state exits the domain, is studied.The players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process. We show that the game has value, and characterize the value function as the unique viscosity solution of an inhomogeneous infinity Laplace equation.Joint work with R. Atar.
Thursday, September 16, 2010 - 15:05 , Location: Skiles 002 , Vladimir Koltchinskii , School of Mathematics, Georgia Tech , Organizer:
We study a problem of estimation of a large Hermitian nonnegatively definite matrix S of unit trace based on n independent measurements                 Y_j = tr(SX_j ) + Z_j , j = 1, . . . , n, where {X_j} are i.i.d. Hermitian matrices and {Z_j } are i.i.d. mean zero random variables independent of {X_j}. Problems of this nature are of interest in quantum state tomography, where S is an unknown density matrix of a quantum system. The estimator is based on penalized least squares method with complexity penalty defined in terms of von Neumann entropy. We derive oracle inequalities showing how the estimation error depends on the accuracy of approximation of the unknown state S by low-rank matrices. We will discuss these results as well as some of the tools used in their proofs (such as generic chaining bounds for empirical processes and noncommutative Bernstein type inequalities).