Sparse canonical correlation analysis from a predictive point of view

Canonical correlation analysis (CCA) describes the associations between two sets of variables by maximizing the correlation between linear combinations of the variables in each data set. However, in high-dimensional settings where the number of variables exceeds the sample size or when the variables are highly correlated, traditional CCA is no longer appropriate. This paper proposes a method for sparse CCA. Sparse estimation produces linear combinations of only a subset of variables from each data set, thereby increasing the interpretability of the canonical variates. We consider the CCA problem from a predictive point of view and recast it into a regression framework. By combining an alternating regression approach together with a lasso penalty, we induce sparsity in the canonical vectors. We compare the performance with other sparse CCA techniques in different simulation settings and illustrate its usefulness on a genomic data set

Sparse multinomial kernel discriminant analysis (sMKDA)

Dimensionality reduction via canonical variate analysis (CVA) is important for pattern recognition and has been extended variously to permit more flexibility, e.g. by "kernelizing" the formulation. This can lead to over-fitting, usually ameliorated by regularization. Here, a method for sparse, multinomial kernel discriminant analysis (sMKDA) is proposed, using a sparse basis to control complexity. It is based on the connection between CVA and least-squares, and uses forward selection via orthogonal least-squares to approximate a basis, generalizing a similar approach for binomial problems. Classification can be performed directly via minimum Mahalanobis distance in the canonical variates. sMKDA achieves state-of-the-art performance in terms of accuracy and sparseness on 11 benchmark datasets

Penalized Orthogonal Iteration for Sparse Estimation of Generalized Eigenvalue Problem

We propose a new algorithm for sparse estimation of eigenvectors in generalized eigenvalue problems (GEP). The GEP arises in a number of modern data-analytic situations and statistical methods, including principal component analysis (PCA), multiclass linear discriminant analysis (LDA), canonical correlation analysis (CCA), sufficient dimension reduction (SDR) and invariant co-ordinate selection. We propose to modify the standard generalized orthogonal iteration with a sparsity-inducing penalty for the eigenvectors. To achieve this goal, we generalize the equation-solving step of orthogonal iteration to a penalized convex optimization problem. The resulting algorithm, called penalized orthogonal iteration, provides accurate estimation of the true eigenspace, when it is sparse. Also proposed is a computationally more efficient alternative, which works well for PCA and LDA problems. Numerical studies reveal that the proposed algorithms are competitive, and that our tuning procedure works well. We demonstrate applications of the proposed algorithm to obtain sparse estimates for PCA, multiclass LDA, CCA and SDR. Supplementary materials are available online

Robust Sparse Canonical Correlation Analysis

Canonical correlation analysis (CCA) is a multivariate statistical method which describes the associations between two sets of variables. The objective is to find linear combinations of the variables in each data set having maximal correlation. This paper discusses a method for Robust Sparse CCA. Sparse estimation produces canonical vectors with some of their elements estimated as exactly zero. As such, their interpretability is improved. We also robustify the method such that it can cope with outliers in the data. To estimate the canonical vectors, we convert the CCA problem into an alternating regression framework, and use the sparse Least Trimmed Squares estimator. We illustrate the good performance of the Robust Sparse CCA method in several simulation studies and two real data examples

Fast Multi-Task SCCA Learning with Feature Selection for Multi-Modal Brain Imaging Genetics

Brain imaging genetics studies the genetic basis of brain structures and functions via integrating both genotypic data such as single nucleotide polymorphism (SNP) and imaging quantitative traits (QTs). In this area, both multi-task learning (MTL) and sparse canonical correlation analysis (SCCA) methods are widely used since they are superior to those independent and pairwise univariate analyses. MTL methods generally incorporate a few of QTs and are not designed for feature selection from a large number of QTs; while existing SCCA methods typically employ only one modality of QTs to study its association with SNPs. Both MTL and SCCA encounter computational challenges as the number of SNPs increases. In this paper, combining the merits of MTL and SCCA, we propose a novel multi-task SCCA (MTSCCA) learning framework to identify bi-multivariate associations between SNPs and multi-modal imaging QTs. MTSCCA could make use of the complementary information carried by different imaging modalities. Using the G2,1-norm regularization, MTSCCA treats all SNPs in the same group together to enforce sparsity at the group level. The l2,1-norm penalty is used to jointly select features across multiple tasks for SNPs, and across multiple modalities for QTs. A fast optimization algorithm is proposed using the grouping information of SNPs. Compared with conventional SCCA methods, MTSCCA obtains improved performance regarding both correlation coefficients and canonical weights patterns. In addition, our method runs very fast and is easy-to-implement, and thus could provide a powerful tool for genome-wide brain-wide imaging genetic studies