Dynamical systems for principal and minor component analysis
Jonathan H. Manton, Uwe Helmke, Iven Mareels · 2004
Principal (PCA) and minor (MCA) component flows are matrix differential equations that converge to the eigenvectors associated with the largest and smallest eigenvalues, respectively, of a given symmetric matrix. They are a useful tool for regression analysis and signal processing, such as e.g. for adaptive antenna arrays, data compression, multi-user detection in wireless communication, and truncated model reduction tasks. Known PCA flows from neural networks include those by Oja, Sanger, Xu, Amari and others. They are closely related also to Brockett’s double bracket flow, but have the advantage that they do not need any initial normalizations of the starting points. These known flows further require the given symmetric matrix to be positive definite and, moroever, finding minor component flows appeared to be harder and unrelated to the principal component case. In this paper we propose a one parameter family of gradient flows that can serve for principal and minor component analysis alike, therefore overcoming previous drawbacks of the theory.