Searcharxiv⌕ Search

arXiv subjects

Ming Yuan

Publications and source records attributed to Ming Yuan.

At least 91 records · Page 5Linked to original sources
arXiv2014

Distance Shrinkage and Euclidean Embedding via Regularized Kernel Estimation

Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the so-called regularized kernel estimate. We show that such an estimate can be characterized as simply applying a constant amount of shrinkage to all observed pairwise distances. This fact allows us to establish risk bounds for the estimate implying that the true distances can be estimated consistently in an average sense as the number of objects increases. In addition, such a characterization suggests an efficient algorithm to compute the distance matrix estimator, as an alternative to the usual second order cone programming known not to scale well for large problems. Numerical experiments and an application in visualizing the diversity of Vpu protein sequences from a recent HIV-1 study further demonstrate the practical merits of the proposed method.

stat.ML↗
arXiv2014

Rate-Optimal Detection of Very Short Signal Segments

Motivated by a range of applications in engineering and genomics, we consider in this paper detection of very short signal segments in three settings: signals with known shape, arbitrary signals, and smooth signals. Optimal rates of detection are established for the three cases and rate-optimal detectors are constructed. The detectors are easily implementable and are based on scanning with linear and quadratic statistics. Our analysis reveals both similarities and differences in the strategy and fundamental difficulty of detection among these three settings.

stat.ML↗
arXiv2014

On Tensor Completion via Nuclear Norm Minimization

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors into matrices and then apply techniques for matrix completion. We show here that such matricization fails to exploit the tensor structure and may lead to suboptimal procedure. More specifically, we investigate a convex optimization approach to tensor completion by directly minimizing a tensor nuclear norm and prove that this leads to an improved sample size requirement. To establish our results, we develop a series of algebraic and probabilistic techniques such as characterization of subdifferetial for tensor nuclear norm and concentration inequalities for tensor martingales, which may be of independent interests and could be useful in other tensor related problems.

stat.ML↗
arXiv2012

Sparsity in multiple kernel learning

The problem of multiple kernel learning based on penalized empirical risk minimization is discussed. The complexity penalty is determined jointly by the empirical $L_2$ norms and the reproducing kernel Hilbert space (RKHS) norms induced by the kernels with a data-driven choice of regularization parameters. The main focus is on the case when the total number of kernels is large, but only a relatively small number of them is needed to represent the target function, so that the problem is sparse. The goal is to establish oracle inequalities for the excess risk of the resulting prediction rule showing that the method is adaptive both to the unknown design distribution and to the sparsity of the problem.

math.ST↗
arXiv2012

A reproducing kernel Hilbert space approach to functional linear regression

We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive definite kernels, we obtain shaper results on the minimax rates of convergence and show that smoothness regularized estimators achieve the optimal rates of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature. Despite the generality of the method of regularization, we show that the procedure is easily implementable. Numerical results are obtained to illustrate the merits of the method and to demonstrate the theoretical developments.

math.ST↗
arXiv2012

Adaptive covariance matrix estimation through block thresholding

Estimation of large covariance matrices has drawn considerable recent attention, and the theoretical focus so far has mainly been on developing a minimax theory over a fixed parameter space. In this paper, we consider adaptive covariance matrix estimation where the goal is to construct a single procedure which is minimax rate optimal simultaneously over each parameter space in a large collection. A fully data-driven block thresholding estimator is proposed. The estimator is constructed by carefully dividing the sample covariance matrix into blocks and then simultaneously estimating the entries in a block by thresholding. The estimator is shown to be optimally rate adaptive over a wide range of bandable covariance matrices. A simulation study is carried out and shows that the block thresholding estimator performs well numerically. Some of the technical tools developed in this paper can also be of independent interest.

math.ST↗
arXiv2012

High Dimensional Semiparametric Gaussian Copula Graphical Models

In this paper, we propose a semiparametric approach, named nonparanormal skeptic, for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider Gaussian Copula graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This celebrating result suggests that the Gaussian copula graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare different estimators for their graph recovery performance under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic dataset to illustrate their empirical usefulness. The R language software package huge implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran. r-project.org/.

stat.ML↗
arXiv2012

The Nonparanormal SKEPTIC

We propose a semiparametric approach, named nonparanormal skeptic, for estimating high dimensional undirected graphical models. In terms of modeling, we consider the nonparanormal family proposed by Liu et al (2009). In terms of estimation, we exploit nonparametric rank-based correlation coefficient estimators including the Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This result suggests that the nonparanormal graphical models are a safe replacement of the Gaussian graphical models, even when the data are Gaussian.

stat.ME↗
arXiv2012

Independent component analysis via nonparametric maximum likelihood estimation

Independent Component Analysis (ICA) models are very popular semiparametric models in which we observe independent copies of a random vector $X = AS$, where $A$ is a non-singular matrix and $S$ has independent components. We propose a new way of estimating the unmixing matrix $W = A^{-1}$ and the marginal distributions of the components of $S$ using nonparametric maximum likelihood. Specifically, we study the projection of the empirical distribution onto the subset of ICA distributions having log-concave marginals. We show that, from the point of view of estimating the unmixing matrix, it makes no difference whether or not the log-concavity is correctly specified. The approach is further justified by both theoretical results and a simulation study.

math.ST↗
arXiv2012

Optimal estimation of the mean function based on discretely sampled functional data: Phase transition

The problem of estimating the mean of random functions based on discretely sampled data arises naturally in functional data analysis. In this paper, we study optimal estimation of the mean function under both common and independent designs. Minimax rates of convergence are established and easily implementable rate-optimal estimators are introduced. The analysis reveals interesting and different phase transition phenomena in the two cases. Under the common design, the sampling frequency solely determines the optimal rate of convergence when it is relatively small and the sampling frequency has no effect on the optimal rate when it is large. On the other hand, under the independent design, the optimal rate of convergence is determined jointly by the sampling frequency and the number of curves when the sampling frequency is relatively small. When it is large, the sampling frequency has no effect on the optimal rate. Another interesting contrast between the two settings is that smoothing is necessary under the independent design, while, somewhat surprisingly, it is not essential under the common design.

math.ST↗
arXiv2012

Support vector machines with a reject option

This paper studies $\ell_1$ regularization with high-dimensional features for support vector machines with a built-in reject option (meaning that the decision of classifying an observation can be withheld at a cost lower than that of misclassification). The procedure can be conveniently implemented as a linear program and computed using standard software. We prove that the minimizer of the penalized population risk favors sparse solutions and show that the behavior of the empirical risk minimizer mimics that of the population risk minimizer. We also introduce a notion of classification complexity and prove that our minimizers adapt to the unknown complexity. Using a novel oracle inequality for the excess risk, we identify situations where fast rates of convergence occur.

math.ST↗
arXiv2010

Structured variable selection and estimation

In linear regression problems with related predictors, it is desirable to do variable selection and estimation by maintaining the hierarchical or structural relationships among predictors. In this paper we propose non-negative garrote methods that can naturally incorporate such relationships defined through effect heredity principles or marginality principles. We show that the methods are very easy to compute and enjoy nice theoretical properties. We also show that the methods can be easily extended to deal with more general regression problems such as generalized linear models. Simulations and real examples are used to illustrate the merits of the proposed methods.

stat.AP↗
arXiv2009

State price density estimation via nonparametric mixtures

We consider nonparametric estimation of the state price density encapsulated in option prices. Unlike usual density estimation problems, we only observe option prices and their corresponding strike prices rather than samples from the state price density. We propose to model the state price density directly with a nonparametric mixture and estimate it using least squares. We show that although the minimization is taken over an infinitely dimensional function space, the minimizer always admits a finite dimensional representation and can be computed efficiently. We also prove that the proposed estimate of the state price density function converges to the truth at a ``nearly parametric'' rate.

q-fin.CP↗
arXiv2009

Convex Optimization Methods for Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression

In this paper, we study convex optimization methods for computing the trace norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection (FES) method, recently proposed by Yuan et al. [22], conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and finite samples. To compute the estimates, however, can be very challenging in practice because of the high dimensionality and the trace norm constraint. In this paper, we explore a variant of Nesterov's smooth method [20] and interior point methods for computing the penalized least squares estimate. The performance of these methods is then compared using a set of randomly generated instances. We show that the variant of Nesterov's smooth method [20] generally outperforms the interior point method implemented in SDPT3 version 4.0 (beta) [19] substantially . Moreover, the former method is much more memory efficient.

stat.ME↗
arXiv2008

Composite quantile regression and the oracle Model Selection Theory

Coefficient estimation and variable selection in multiple linear regression is routinely done in the (penalized) least squares (LS) framework. The concept of model selection oracle introduced by Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] characterizes the optimal behavior of a model selection procedure. However, the least-squares oracle theory breaks down if the error variance is infinite. In the current paper we propose a new regression method called composite quantile regression (CQR). We show that the oracle model selection theory using the CQR oracle works beautifully even when the error variance is infinite. We develop a new oracular procedure to achieve the optimal properties of the CQR oracle. When the error variance is finite, CQR still enjoys great advantages in terms of estimation efficiency. We show that the relative efficiency of CQR compared to the least squares is greater than 70% regardless the error distribution. Moreover, CQR could be much more efficient and sometimes arbitrarily more efficient than the least squares. The same conclusions hold when comparing a CQR-oracular estimator with a LS-oracular estimator.

math.ST↗
arXiv2008

Structured variable selection in support vector machines

When applying the support vector machine (SVM) to high-dimensional classification problems, we often impose a sparse structure in the SVM to eliminate the influences of the irrelevant predictors. The lasso and other variable selection techniques have been successfully used in the SVM to perform automatic variable selection. In some problems, there is a natural hierarchical structure among the variables. Thus, in order to have an interpretable SVM classifier, it is important to respect the heredity principle when enforcing the sparsity in the SVM. Many variable selection methods, however, do not respect the heredity principle. In this paper we enforce both sparsity and the heredity principle in the SVM by using the so-called structured variable selection (SVS) framework originally proposed in Yuan, Joseph and Zou (2007). We minimize the empirical hinge loss under a set of linear inequality constraints and a lasso-type penalty. The solution always obeys the desired heredity principle and enjoys sparsity. The new SVM classifier can be efficiently fitted, because the optimization problem is a linear program. Another contribution of this work is to present a nonparametric extension of the SVS framework, and we propose nonparametric heredity SVMs. Simulated and real data are used to illustrate the merits of the proposed method.

stat.ML↗