发表于： 2012-05-29 13:51 点击：651

** Title:** High-Dimensional Statistical Inference: From Vectors to Matrices

** Speaker: ** T. Tony Cai, Department of Statisticsm, University of Pennsylvania

**Abstract: ** Driven by a wide range of applications, high-dimensional statistical inference has seen significant developments over the last few years. These and other related problems have also attracted much interest in a number of fields including applied mathematics, engineering, and statistics. In this talk I will discuss some recent advances on several problems in high-dimensional inference including compressed sensing, low-rank matrix recovery, and estimation of large covariance matrices. The connections as well as differences among these problems will be also discussed.

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** Title:** ESTIMATION OF SPIKED EIGENVALUES IN SPIKED MODELS

** Speaker: ** Xue Ding, School of Mathematics, Jilin University

**Abstract: ** The paper proposes new estimators of spiked eigenvalues of the population covariance matrix from the sample covariance matrix and investigates its consistency and asymptotic normality.

** Title:** Distributions of Eigenvalues of Large Euclidean Matrices Generated from Three Manifolds

** Speaker: ** Tiefeng Jiang, School of Statistics, University of Minnesota at Twin Cities

**Abstract: ** Many problems on Euclidean random matrices appeared in different disciplines including Physics, Mathematics and Statistics. Choose n random points from a set G randomly and uniformly, where G is a p-dimensional sphere, ball or cube. The n by n Euclidean random matrix is defined by M where the (i,j) entry of M is a function f of the Euclidean distance between the ith and jth points. We show that, F(n), the empirical distribution of the eigenvalues of M, converges to the point mass at 0 in distribution as n goes to infinity for fixed p. When p and n are proportional to each other, we show that F(n) converges weakly to the distribution of a + bV, where V is has the Marcenko-Pastur law. Taking a special function of f, we obtain the eigenvalue behavior of the non-Euclidean matrix. Many examples related to Physics are also given.

** Title:** Circular Law and Arc Law for Truncation of Random Unitary Matrices

** Speaker: ** Danning Li, School of Statistics, University of Minnesota at Twin Cities

**Abstract: ** Let V be the m x m upper-left corner of an n x n Haar-invariant unitary matrix. Let a_1,..., a_m be the eigenvalues of V. We prove that the empirical distribution of a normalization of a_1,..., a_m goes to the circular law as m ->infty with m/n -> 0. We also prove that the empirical distribution of a_1,..., a_m goes to the arc law as m/n -> 1. This is a joint work with Zhishan Dong and Tiefeng Jiang.

** Title:** Sparse Singular Value Decomposition in High Dimensions

** Speaker: ** Zongming Ma, The Wharton School, University of Pennsylvania

**Abstract: ** Singular value decomposition is a widely used tool for dimension reduction in ultivariate analysis. However, when used for statistical estimation in high-dimensional low rank matrix models, singular vectors of the noise-corrupted matrix are inconsistent for their counterparts of the true mean matrix. In this talk, we suppose the true singular vectors have sparse representations in a certain basis. We propose an iterative thresholding algorithm that can estimate the subspaces spanned by leading left and right singular vectors and also the true mean matrix optimally under Gaussian assumption. We further turn the algorithm into a practical methodology that is fast, data-driven and robust to heavy-tailed noises. Simulations and a real data example further show its competitive performance. This is a joint work with Andreas Buja and Dan Yang.

** Title:** A Markov Chain for Covariate-Adaptive Biased Coin Randomization

** Speaker: ** Lixin Zhang, Institute of Statistics, Zhejiang University

**Abstract: ** Stratified permuted block randomization and marginal method (Pocock and Simon, 1975) are the two most popular approaches for balancing treatment allocation for influential covariates in clinical trials. Stratified permuted block randomization achieves good balance when the number of strata is small and marginal procedure achieves good within-covariate-margin balance. Simulation studies show that they have large imbalances either as a whole, or within-stratum when the number of strata is large. Hu and Hu (2012) proposed a new randomization that gives a good tradeoff between them by minimizing a weighted average of with-stratum, within-covariate-margin and overall imbalances. However, the theoretical results are established under a very strict condition on the weights which is hard to be satisfied in practices and do not apply to Pocock and Simion's design whose theoretical bases remain largely elusive for decades.

In this talk, we will give the theory of a general framework of covariate-adaptive randomization under widely satisfied condition by developing the theory of a high-dimensional Markov chain. In particular, we verify in theory that under Pocock and Simon's marginal procedure, the within-covariate-margin imbalances and overall imbalances are bounded in probability, but the within-stratum imbalances increase with the rate of $\sqrt{n}$ as the sample size increases. Our theory gives both a critical condition for the covariate-adaptive randomization to achieve good within-stratum balance and a critical condition to ac