Sparse Variable PCA Using Geodesic Steepest Descent

Principal component analysis (PCA) is a dimensionality reduction technique used in most fields of science and engineering. It aims to find linear combinations of the input variables that maximize variance. A problem with PCA is that it typically assigns nonzero loadings to all the variables, which i...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2008-12, Vol.56 (12), p.5823-5832
Main Authors: Ulfarsson, M.O., Solo, V.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Bayesian methods
Biological and medical sciences
Computer simulation
Computerized, statistical medical data processing and models in biomedicine
Covariance matrix
Criteria
Descent
Exact sciences and technology
Geodesic
geometric
Information, signal and communications theory
Input variables
l_{1}
LASSO
Magnetic analysis
Magnetic resonance imaging
Mathematical models
Medical management aid. Diagnosis aid
Medical sciences
Miscellaneous
Optimization
Personal communication networks
Principal component analysis
principal component analysis (PCA)
Reduction
regularization
Signal processing
Signal processing algorithms
sparse
Sparse matrices
Studies
Telecommunications and information theory
Tuning
Variables
Vectors
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Summary:Principal component analysis (PCA) is a dimensionality reduction technique used in most fields of science and engineering. It aims to find linear combinations of the input variables that maximize variance. A problem with PCA is that it typically assigns nonzero loadings to all the variables, which in high dimensional problems can require a very large number of coefficients. But in many applications, the aim is to obtain a massive reduction in the number of coefficients. There are two very different types of sparse PCA problems: sparse loadings PCA (slPCA) which zeros out loadings (while generally keeping all of the variables) and sparse variable PCA which zeros out whole variables (typically leaving less than half of them). In this paper, we propose a new svPCA, which we call sparse variable noisy PCA (svnPCA). It is based on a statistical model, and this gives access to a range of modeling and inferential tools. Estimation is based on optimizing a novel penalized log-likelihood able to zero out whole variables rather than just some loadings. The estimation algorithm is based on the geodesic steepest descent algorithm. Finally, we develop a novel form of Bayesian information criterion (BIC) for tuning parameter selection. The svnPCA algorithm is applied to both simulated data and real functional magnetic resonance imaging (fMRI) data.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2008.2006587