Journal
BIOSTATISTICS
Volume 10, Issue 3, Pages 515-534Publisher
OXFORD UNIV PRESS
DOI: 10.1093/biostatistics/kxp008
Keywords
Canonical correlation analysis; DNA copy number; Integrative genomic analysis; L-1; Matrix decomposition; Principal component analysis; Sparse principal component analysis; SVD
Funding
- NCI NIH HHS [2R01 CA 72028-07] Funding Source: Medline
- NHLBI NIH HHS [N01-HV-28183] Funding Source: Medline
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We present a penalized matrix decomposition (PMD), a new framework for computing a rank-K approximation for a matrix. We approximate the matrix X as (X) over cap = Sigma(K)(k=1) d(k)u(k)V(k)(T,) where d(k), u(k), and v(k) minimize the squared Frobenius norm of X - (X) over cap, subject to penalties on u(k) and v(k). This results in a regularized version of the singular value decomposition. Of particular interest is the use of L-1-penalties on u(k) and v(k), which yields a decomposition of X using sparse vectors. We show that when the PMD is applied using an L-1-penalty on v(k) but not on u(k), a method for sparse principal components results. In fact, this yields an efficient algorithm for the SCoTLASS proposal (Jolliffe and others 2003) for obtaining sparse principal components. This method is demonstrated on a publicly available gene expression data set. We also establish connections between the SCoTLASS method for sparse principal component analysis and the method of Zou and others (2006). In addition, we show that when the PMD is applied to a cross-products matrix, it results in a method for penalized canonical correlation analysis (CCA). We apply this penalized CCA method to simulated data and to a genomic data set consisting of gene expression and DNA copy number measurements on the same set of samples.
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