Oja’s algorithm for graph clustering, Markov spectral decomposition, and risk sensitive control

• Published in: Automatica (Oxford) Vol. 48; no. 10; pp. 2512 - 2519
• Main Authors: Borkar, V., Meyn, S.P.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.10.2012; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Computer science; control theory; systems; Data processing. List processing. Character string processing; Exact sciences and technology; Graph algorithms; Information retrieval. Graph; Markov chains; Mathematics; Memory organisation. Data processing; Multiplicative ergodic theory; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Oja’s algorithm; Risk sensitive control; Sciences and techniques of general use; Software; Spectral theory of Markov chains; Stochastic approximation; Theoretical computing; Stochastic approximation; Markov chains; Multiplicative ergodic theory; Graph algorithms; Risk sensitive control; Spectral theory of Markov chains; Oja’s algorithm; Cluster analysis; Ergodic theory; Multiplication; Modeling; Markov chain; Graph decomposition; Vector space; Graph connectivity; Operator splitting; Eigenvalue problem; Stationary process; Oja's algorithm; Spectral method; Factor analysis; Statistical analysis; Probabilistic approach; Eigenvector; Matrix decomposition; Stationary condition; Robust control; Spectral theory; Positive definite matrix; Principal component analysis
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2012.05.016

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. Recall that this means that the stationary distribution satisfies the detailed balance equations (Meyn & Tweedie, 2009). Equivalently, the statistics of the process in steady state do not change when time is reversed. Stability and convergence of Oja’s algorithm are established under conditions far milder than that assumed in previous work. Applications to graph clustering, Markov spectral decomposition, and multiplicative ergodic theory are surveyed, along with numerical results.