Multiple Response Regression for Gaussian Mixture Models with Known Labels

Double Sparsity Kernel Learning with Automatic Variable Selection and Data Extraction

Learning in the Reproducing Kernel Hilbert Space (RKHS) has been widely used in many scientific disciplines. Because a RKHS can be very flexible, it is common to impose a regularization term in the optimization to prevent overfitting. Standard RKHS learning employs the squared norm penalty of the learning function. Despite its success, many challenges remain. In particular, one cannot directly use the squared norm penalty for variable selection or data extraction. Therefore, when there exists noise predictors, or the underlying function has a sparse representation in the dual space, the performance of standard RKHS learning can be suboptimal. In the literature, work has been proposed on how to perform variable selection in RKHS learning, and a data sparsity constraint was considered for data extraction. However, how to learn in a RKHS with both variable selection and data extraction simultaneously remains unclear. In this paper, we propose a unified RKHS learning method, namely, DOuble Sparsity Kernel (DOSK) learning, to overcome this challenge. An efficient algorithm is provided to solve the corresponding optimization problem. We prove that under certain conditions, our new method can asymptotically achieve variable selection consistency. Simulated and real data results demonstrate that DOSK is highly competitive among existing approaches for RKHS learning.

Joint Estimation of Multiple Precision Matrices with Common Structures

Estimation of inverse covariance matrices, known as precision matrices, is important in various areas of statistical analysis. In this article, we consider estimation of multiple precision matrices sharing some common structures. In this setting, estimating each precision matrix separately can be suboptimal as it ignores potential common structures. This article proposes a new approach to parameterize each precision matrix as a sum of common and unique components and estimate multiple precision matrices in a constrained l₁ minimization framework. We establish both estimation and selection consistency of the proposed estimator in the high dimensional setting. The proposed estimator achieves a faster convergence rate for the common structure in certain cases. Our numerical examples demonstrate that our new estimator can perform better than several existing methods in terms of the entropy loss and Frobenius loss. An application to a glioblastoma cancer data set reveals some interesting gene networks across multiple cancer subtypes.