Yiyuan She

Optimal selection of reduced rank estimators of high-dimensional matrices

We introduce a new criterion, the Rank Selection Criterion (RSC), for selecting the optimal reduced rank estimator of the coefficient matrix in multivariate response regression models. The corresponding RSC estimator minimizes the Frobenius norm of the fit plus a regularization term proportional to the number of parameters in the reduced rank model. The rank of the RSC estimator provides a consistent estimator of the rank of the coefficient matrix; in general, the rank of our estimator is a consistent estimate of the effective rank, which we define to be the number of singular values of the target matrix that are appropriately large. The consistency results are valid not only in the classic asymptotic regime, when n, the number of responses, and p, the number of predictors, stay bounded, and m, the number of observations, grows, but also when either, or both, n and p grow, possibly much faster than m. We establish minimax optimal bounds on the mean squared errors of our estimators. Our finite sample performance bounds for the RSC estimator show that it achieves the optimal balance between the approximation error and the penalty term. Furthermore, our procedure has very low computational complexity, linear in the number of candidate models, making it particularly appealing for large scale problems. We contrast our estimator with the nuclear norm penalized least squares (NNP) estimator, which has an inherently higher computational complexity than RSC, for multivariate regression models. We show that NNP has estimation properties similar to those of RSC, albeit under stronger conditions. However, it is not as parsimonious as RSC. We offer a simple correction of the NNP estimator which leads to consistent rank estimation. We verify and illustrate our theoretical findings via an extensive simulation study.

Outlier Detection Using Nonconvex Penalized Regression

This article studies the outlier detection problem from the standpoint of penalized regression. In the regression model, we add one mean shift parameter for each of the n data points. We then apply a regularization favoring a sparse vector of mean shift parameters. The usual L1 penalty yields a convex criterion, but fails to deliver a robust estimator. The L1 penalty corresponds to soft thresholding. We introduce a thresholding (denoted by Θ) based iterative procedure for outlier detection (Θ–IPOD). A version based on hard thresholding correctly identifies outliers on some hard test problems. We describe the connection between Θ–IPOD and M-estimators. Our proposed method has one tuning parameter with which to both identify outliers and estimate regression coefficients. A data-dependent choice can be made based on the Bayes information criterion. The tuned Θ–IPOD shows outstanding performance in identifying outliers in various situations compared with other existing approaches. In addition, Θ–IPOD is much faster than iteratively reweighted least squares for large data, because each iteration costs at most O(np) (and sometimes much less), avoiding an O(np2) least squares estimate. This methodology can be extended to high-dimensional modeling with p ≫ n if both the coefficient vector and the outlier pattern are sparse.

Thresholding-based iterative selection procedures for model selection and shrinkage

This paper discusses a class of thresholding-based iterative selection procedures (TISP) for model selection and shrinkage. People have long before noticed the weakness of the convex

Joint variable and rank selection for parsimonious estimation of high-dimensional matrices

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The Group Square-Root Lasso: Theoretical Properties and Fast Algorithms

-constraint (or the soft-thresholding) in wavelets and have designed many different forms of nonconvex penalties to increase model sparsity and accuracy. But for a nonorthogonal regression matrix, there is great difficulty in both investigating the performance in theory and solving the problem in computation. TISP provides a simple and efficient way to tackle this so that we successfully borrow the rich results in the orthogonal design to solve the nonconvex penalized regression for a general design matrix. Our starting point is, however, thresholding rules rather than penalty functions. Indeed, there is a universal connection between them. But a drawback of the latter is its non-unique form, and our approach greatly facilitates the computation and the analysis. In fact, we are able to build the convergence theorem and explore theoretical properties of the selection and estimation via TISP nonasymptotically. More importantly, a novel Hybrid-TISP is proposed based on hard-thresholding and ridge-thresholding. It provides a fusion between the

Why Deep Learning Works: A Manifold Disentanglement Perspective

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Feature Selection with Annealing for Computer Vision and Big Data Learning

-penalty and the

Group Iterative Spectrum Thresholding for Super-Resolution Sparse Spectral Selection

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Matrix Factorizations for Parallel Integer Transformation

-penalty, and adaptively achieves the right balance between shrinkage and selection in statistical modeling. In practice, Hybrid-TISP shows superior performance in test-error and is parsimonious.

Multivariate calibration maintenance and transfer through robust fused LASSO

We propose dimension reduction methods for sparse, high-dimensional multivariate response regression models. Both the number of responses and that of the predictors may exceed the sample size. Sometimes viewed as complementary, predictor selection and rank reduction are the most popular strategies for obtaining lower-dimensional approximations of the parameter matrix in such models. We show in this article that important gains in prediction accuracy can be obtained by considering them jointly. We motivate a new class of sparse multivariate regression models, in which the coefficient matrix has low rank and zero rows or can be well approximated by such a matrix. Next, we introduce estimators that are based on penalized least squares, with novel penalties that impose simultaneous row and rank restrictions on the coefficient matrix. We prove that these estimators indeed adapt to the unknown matrix sparsity and have fast rates of convergence. We support our theoretical results with an extensive simulation study and two data analyses.