## Seminars and Colloquia by Series

Thursday, April 7, 2011 - 15:05 , Location: Skiles 005 , Heinrich Matzinger , Georgia Tech , , Organizer: Heinrich Matzinger
We consider two random sequences of equal length n and the alignments with gaps corresponding to their Longest Common Subsequences. These alignments are called optimal alignments. What are the properties of these alignments? What are the proportion of different aligned letter pairs? Are there concentration of measure properties for these proportions? We will see that the convex geometry of the asymptotic limit set of empirical distributions seen along alignments can determine the answer to the above questions.
Thursday, March 31, 2011 - 15:05 , Location: Skiles 005 , Jan Rosinski , University of Tennessee, Knoxville , Organizer:
Semimartingales constitute the larges class of "good integrators" for which Ito integral could reasonably be defined and the stochastic analysis machinery applied. In this talk we identify semimartingales within certain infinitely divisible processes. Examples include stationary (but not independent) increment processes, such as fractional and moving average processes, as well as their mixtures. Such processes are non-Markovian, often possess long range memory, and are of interest as stochastic integrators. The talk is based on a joint work with Andreas Basse-O'Connor.
Thursday, March 10, 2011 - 15:05 , Location: Skiles 005 , Jonathan Mattingly , Duke University, Mathematics Department , Organizer:
I will discuss how the idea of coupling at time infinity is equivalent to unique ergodicity of a markov process. In general, the coupling will be a kind of "asymptotic Wasserstein" coupling.  I will draw examples from SDEs with memory and SPDEs. The fact that both are infinite dimensional markov processes is no coincidence.
Thursday, March 3, 2011 - 15:05 , Location: Skiles 005 , Stas Minsker , Georgia Tech , Organizer:
Let (X,Y) be a random couple with unknown distribution P, X being an observation and Y - a binary label to be predicted. In practice, distribution P remains unknown but the learning algorithm has access to the training data - the sample from P. It often happens that the cost of obtaining the training data is associated with labeling the observations while the pool of observations itself is almost unlimited. This suggests to measure the performance of a learning algorithm in terms of its label complexity, the number of labels required to obtain a classifier with the desired accuracy. Active Learning theory explores the possible advantages of this modified framework.We will present a new active learning algorithm based on nonparametric estimators of the regression function and explain main improvements over the previous work.Our investigation provides upper and lower bounds for the performance of proposed method over a broad class of underlying distributions.
Tuesday, March 1, 2011 - 16:05 , Location: Skiles 006 , Robert Ziff , Michigan Center for Theoretical Physics, Department of Chemical Engineering, University of Michigan , Organizer:
Various exact results in two-dimensional percolation are presented. A method for finding exact thresholds for a wide variety of systems, which greatly expands previously known exactly solvable systems to such new lattices as "martini" and generalized "bowtie" lattices, is given. The size distribution is written in a Zipf's-law form in terms of the enclosed- area distribution, and the coefficient can be written in terms of the the number of hulls crossing a cylinder.  Additional properties of hull walks (equivalent to some kinds of trajectories) are given.  Finally, some ratios of correlation functions are shown to be universal, with a functional form that can be found exactly from conformal field theory.
Thursday, February 24, 2011 - 15:05 , Location: Skyles 005 , , Computer Sciences Department, University of Wisconsin , Organizer:
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 , , University of California, San Diego , Organizer:
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