Matrix factorization methods are extensively employed to understand complex data. In this paper, we introduce the cross-product penalized component analysis (XCAN), a sparse matrix factorization based on the optimization of a loss function that allows a trade-off between variance maximization and structural preservation. The approach is based on previous developments, notably (i) the Sparse Principal Component Analysis (SPCA) framework based on the LASSO, (ii) extensions of SPCA to constrain both modes of the factorization, like co-clustering or the Penalized Matrix Decomposition (PMD), and (iii) the Group-wise Principal Component Analysis (GPCA) method. The result is a flexible modeling approach that can be used for data exploration in a large variety of problems. We demonstrate its use with applications from different disciplines.

Department of Statistics, Rice University Abstract High-dimensional data common in genomics, proteomics, and chemometrics often contains complicated correlation structures. Recently, partial least squares (PLS) and Sparse PLS methods have gained attention in these areas as dimension reduction techniques in the context of supervised data analysis. We introduce a framework for Regularized PLS by solving a relaxation of the SIMPLS optimization problem with penalties on the PLS loadings vectors. Our approach enjoys many advantages including flexibility, general penalties, easy interpretation of results, and fast computation in high-dimensional settings. We also outline extensions of our methods leading to novel methods for Nonnegative PLS and Generalized PLS, an adaption of PLS for structured data. We demonstrate the utility of our methods through simulations and a case study on proton Nuclear Magnetic Resonance (NMR) spectroscopy data. To whom correspondence should be addressed; Department of Statistics, Rice University, MS 138, 6100 Main St., Houston, TX 77005 (email: gallen@rice.edu) 1 Introduction Technologies to measure high-throughput biomedical data in proteomics, chemometrics, and genomics have led to a proliferation of high-dimensional data that pose many statistical challenges. As genes, proteins, and metabolites, are biologically interconnected, the variables in these data sets are often highly correlated. In this context, several have recently advocated using partial least squares (PLS) for dimension reduction of supervised data, or data with a response or labels (Nguyen and Rocke, 2002b; Boulesteix and Strimmer, 2007; Rossouw et al., 2008; Chun and Keleş, 2010). First introduced by Wold (1966) as a regression method that uses least squares on a set of derived inputs accounting for multi-colinearities, others have since proposed alternative methods for PLS with multiple responses (de Jong, 1993) and for classification (Marx, 1996; Barker and Rayens, 2003).