Recursive Principal Components Analysis Using Eigenvector Matrix Perturbation
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Recursive Principal Components Analysis Using Eigenvector Matrix Perturbation Deniz Erdogmus Department of Computer Science and Engineering, CSE, Oregon Graduate Institute, Oregon Health & Science University, Beaverton, OR 97006, USA Email: [email protected]
Yadunandana N. Rao Computational NeuroEngineering Laboratory (CNEL), Department of Electrical & Computer Engineering (ECE), University of Florida, Gainesville, FL 32611, USA Email: [email protected]
Hemanth Peddaneni Computational NeuroEngineering Laboratory (CNEL), Department of Electrical & Computer Engineering (ECE), University of Florida, Gainesville, FL 32611, USA Email: [email protected]
Anant Hegde Computational NeuroEngineering Laboratory (CNEL), Department of Electrical & Computer Engineering (ECE), University of Florida, Gainesville, FL 32611, USA Email: [email protected]
Jose C. Principe Computational NeuroEngineering Laboratory (CNEL), Department of Electrical & Computer Engineering (ECE), University of Florida, Gainesville, FL 32611, USA Email: [email protected] Received 4 December 2003; Revised 19 March 2004; Recommended for Publication by John Sorensen Principal components analysis is an important and well-studied subject in statistics and signal processing. The literature has an abundance of algorithms for solving this problem, where most of these algorithms could be grouped into one of the following three approaches: adaptation based on Hebbian updates and deflation, optimization of a second-order statistical criterion (like reconstruction error or output variance), and fixed point update rules with deflation. In this paper, we take a completely different approach that avoids deflation and the optimization of a cost function using gradients. The proposed method updates the eigenvector and eigenvalue matrices simultaneously with every new sample such that the estimates approximately track their true values as would be calculated from the current sample estimate of the data covariance matrix. The performance of this algorithm is compared with that of traditional methods like Sanger’s rule and APEX, as well as a structurally similar matrix perturbation-based method. Keywords and phrases: PCA, recursive algorithm, rank-one matrix update.
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INTRODUCTION
Principal components analysis (PCA) is a well-known statistical technique that has been widely applied to solve important signal processing problems like feature extraction, signal estimation, detection, and speech separation [1, 2, 3, 4]. Many analytical techniques exist, which can solve PCA once the entire input data is known [5]. However, most of the analytical methods require extensive matrix operations and
hence they are unsuited for real-time applications. Further, in many applications such as direction of arrival (DOA) tracking, adaptive subspace estimation, and so forth, signal statistics change over time rendering the block methods virtually unacceptable. In such cases, fast, adaptive, on-line solutions are desirable. Majority of the existing algorithms for PCA are based on st
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