Department of Mathematics and Statistics

Friday, January 25th, 2019

Time: 2:30 p.m.  Place: Jeffery Hall 234

Speaker: Qian Qin (University of Florida)

Title: Convergence complexity analysis of MCMC.

Abstract: Convergence complexity analysis is the study of how Markov chain Monte Carlo (MCMC) algorithms used in Bayesian statistics scale with the size of the underlying data set. To conduct this type of analysis, one needs tools to construct convergence bounds for high-dimensional Markov chains. I will review a few classical techniques of Markov chain convergence analysis (in particular, drift and minorization), and discuss their applicability and limitations in high-dimensional settings. I will then present a result concerning the convergence complexity of Albert and Chib's algorithm for Bayesian probit regression.

Qian Qin is a Ph.D. Candidate in Statistics at the University of Florida. His research interests include Markov chain Monte Carlo, Bayesian statistics, and high-dimensional statistics.

Tuesday, January 22nd, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Steven Spallone (Indian Institute of Science Education and Research, Pune)

Title: A Chinese Remainder Theorem for Young Diagrams.

Abstract: Given a natural number 't' and a Young Diagram 'Y', there is a notion of a "remainder of Y upon division by t", called the t-core of Y. Let s,t be relatively prime, and consider the map taking a given st-core Y to the pair consisting of its s-core and t-core. The fibres of this map are infinite. More precisely, we have proven that the cardinality of the set of length k members of a given fibre is a quasipolynomial in k, of degree st-s-t. This is joint work with K. Seethalakshmi.

Monday, January 21st, 2019

Time: 4:30 p.m.  Place: Jeffery Hall 234

Speaker: Michelle Miranda (University of Texas)

Title: Modeling Modern Data Objects: Statistical Methods for Ultra-high Dimensionality and Intricate Correlation Structures.

Abstract: Advances in technology have been generating data with increased complexity. Modern data objects are often high-dimensional and can also lay in 2D, 3D and even 4D Euclidean and sometimes non-Euclidean planes. One example are data generated from large-scale multi-site studies such as the Human Connectome Project and the Alzheimer's Disease Neuroimaging Initiative. In these scenarios, complexity comes in the form of brain images such as functional magnetic resonance imaging (fMRI), a 4D object that is high-dimensional (number of voxels is around 100K for each time point and each subject), with measurements that are correlated both in time and in space. Another example are data coming from an instrument specifically designed to measure scleral strain in donor eyes. The device generates functional data with functions defined on a non-Euclidean 2D partial spherical domain. Associating these complex data with clinical, environmental, and genetic variables can be challenging and classical statistical tools need to evolve to keep up with the complexity of these new data types. In this talk I address some of the challenges brought by modern data objects and show a few solutions to some important statistical questions in this context.

Michelle Miranda obtained her Ph.D. degree at the University of North Carolina at Chapel Hill in 2014. She is currently a Postdoctoral Fellow in Biostatistics at The University of Texas MD Anderson Cancer Center. Her research interests include Bayesian analysis with focus on high-dimensional settings, functional data analysis, and methods for correlated data objects.

Wednesday, January 16th, 2019

Time: 3:30 p.m.  Place: Jeffery Hall 234

Speaker: Long Feng (Yale)

Title: Sorted Concave Penalized Regression.

Abstract: The Lasso is biased. Concave penalized lease squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and estimation, the bias of the Lasso can be also reduced by taking a smaller penalty level than what selection consistency requires, but such smaller penalty level depends on the sparsity of the true coefficient vector. The sorted L1 penalized estimation (Slope) was proposed for adaptation to such smaller penalty levels. However, the advantages of concave PLSE and Slope do not subsume each other. We propose sorted concave penalized estimation to combine the advantages of concave and sorted penalizations. We prove that sorted concave penalties adaptively choose the smaller penalty level and at the same time benefits from signal strength, especially when a significant proportion of signals are stronger than the corresponding adaptively selected penalty levels. A local convex approximation, which extends the local linear and quadratic approximations to sorted concave penalties, is developed to facilitate the computation of sorted concave PLSE and proven to possess desired prediction and estimation error bounds. We carry out a unified treatment of penalty functions in a general optimization setting, including the penalty levels and concavity of the above mentioned sorted penalties and mixed penalties motivated by Bayesian considerations. Our analysis of prediction and estimation errors requires the restricted eigenvalue condition on the design, not beyond, and provides selection consistency under a required minimum signal strength condition in addition. Thus, our results also sharpens existing results on concave PLSE by removing the upper sparse eigenvalue component of the sparse Riesz condition.

Long Feng obtained his Ph.D. degree at the Department of Statistics and Biostatistics, Rutgers University, in 2017. He is currently a Postdoctoral Associate at Yale University. His research interests include high-dimensional statistics, variable selection, empirical Bayes methods, tensor regression, imaging data analysis, and convex/non-convex optimizations.

Tuesday, January 8th, 2019

Time: 1:00-2:00 p.m.  Place: Jeffery Hall 422

Speaker: Anup Dixit (Queen's University)

Title: Large values on the 1-line for a family of L-functions.

Abstract: A classical problem in analytic number theory is to understand the values of L-functions in the critical strip. It is well-known that $|\zeta(1+it)|$ takes arbitrarily large values when $t$ runs through the real numbers. In 2006, Granville and Soundarajan conjectured that there exists arbitrarily large $t$ such that $|\zeta(1+it)|$ satisfies a certain lower bound. We discuss recent progress towards this conjecture and also generalize it to a family of L-functions. This is joint work with K. Mahatab.