A Least-Squares Framework for Component Analysis

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 34; no. 6; pp. 1041 - 1055
• Main Author: De la Torre, F.
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01.06.2012; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithm design and analysis; Algorithms; Analytical models; Applied sciences; canonical correlation analysis; Clustering; Computer science; control theory; systems; Covariance matrix; Data processing. List processing. Character string processing; dimensionality reduction; Discriminant analysis; Equations; Exact sciences and technology; Exact solutions; Humans; Intelligence; k-means; Kernel; kernel methods; Kernels; Least squares method; Least-Squares Analysis; linear discriminant analysis; Mathematical analysis; Mathematical model; Mathematical models; Matrices; Memory organisation. Data processing; Pattern Recognition, Automated - methods; Principal component analysis; Principal Component Analysis - methods; Principal components analysis; reduced rank regression; Sample Size; Software; spectral clustering; Studies; Cluster analysis; Visualization; Sample size; K means algorithm; Potential function; Modeling; Canonical correlation; dimensionality reduction; Least squares method; Classification; k-means; linear discriminant analysis; Pattern extraction; spectral clustering; Discriminant analysis; Spectral method; reduced rank regression; Factor analysis; Locality; Cluster; Regression analysis; Pattern recognition; canonical correlation analysis; Kernel method; Exact solution; Dimension reduction; Problem solving; kernel methods; Feature extraction; Principal component analysis
• ISSN: 0162-8828; 1939-3539; 2160-9292
• DOI: 10.1109/TPAMI.2011.184

Over the last century, Component Analysis (CA) methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Canonical Correlation Analysis (CCA), Locality Preserving Projections (LPP), and Spectral Clustering (SC) have been extensively used as a feature extraction step for modeling, classification, visualization, and clustering. CA techniques are appealing because many can be formulated as eigen-problems, offering great potential for learning linear and nonlinear representations of data in closed-form. However, the eigen-formulation often conceals important analytic and computational drawbacks of CA techniques, such as solving generalized eigen-problems with rank deficient matrices (e.g., small sample size problem), lacking intuitive interpretation of normalization factors, and understanding commonalities and differences between CA methods. This paper proposes a unified least-squares framework to formulate many CA methods. We show how PCA, LDA, CCA, LPP, SC, and its kernel and regularized extensions correspond to a particular instance of least-squares weighted kernel reduced rank regression (LS--WKRRR). The LS-WKRRR formulation of CA methods has several benefits: 1) provides a clean connection between many CA techniques and an intuitive framework to understand normalization factors; 2) yields efficient numerical schemes to solve CA techniques; 3) overcomes the small sample size problem; 4) provides a framework to easily extend CA methods. We derive weighted generalizations of PCA, LDA, SC, and CCA, and several new CA techniques.