Oja's algorithm for graph clustering and Markov spectral decomposition

Authors:
V. Borkar;S. P. Meyn
Affiliations:
Tata Institute of Fundamental Research, Mumbai, India;University of Illinois at Urbana-Champaign, Urbana, IL
Venue:
Proceedings of the 3rd International Conference on Performance Evaluation Methodologies and Tools
Year:
2008

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Abstract

Given a positive definite matrix M and an integer Nm ≥ 1, Oja's subspace algorithm will provide convergent estimates of the first Nm eigenvalues of M along with the corresponding eigenvectors. It is a common approach to principal component analysis. This paper introduces a normalized stochastic-approximation implementation of Oja's subspace algorithm, as well as new applications to the spectral decomposition of a reversible Markov chain. Stability and convergence are established under conditions far milder than assumed in previous work. Applications to graph clustering and Markov spectral decomposition are surveyed, along with numerical results.