Adaptive Normalized Quasi-Newton Algorithms for Extraction of Generalized Eigen-Pairs and Their Convergence Analysis

作者：Tuan Duong Nguyen^*; Yamada Isao

来源：IEEE Transactions on Signal Processing, 2013, 61(6): 1404-1418.

DOI：10.1109/TSP.2012.2234744

• Adaptive algorithm
• deterministic discrete-time (DDT) approach
• generalized Hermitian eigenvalue problem (GHEP)
• minor generalized eigenvector
• Newton's method
• principal generalized eigenvector

The main contributions of this paper are to propose and analyze fast and numerically stable adaptive algorithms for the generalized Hermitian eigenvalue problem (GHEP), which arises in many signal processing applications. First, for given explicit knowledge of a matrix pencil, we formulate two novel deterministic discrete-time (DDT) systems for estimating the generalized eigen-pair (eigenvector and eigenvalue) corresponding to the largest/smallest generalized eigenvalue. By characterizing a generalized eigen-pair as a stationary point of a certain function, the proposed DDT systems can be interpreted as natural combinations of the normalization and quasi-Newton steps for finding the solution. Second, we present adaptive algorithms corresponding to the proposed DDT systems. Moreover, we establish rigorous analysis showing that, for a step size within a certain range, the sequence generated by the DDT systems converges to the orthogonal projection of the initial estimate onto the generalized eigensubspace corresponding to the largest/smallest generalized eigenvalue. Numerical examples demonstrate the practical applicability and efficacy of the proposed adaptive algorithms.

• 出版日期2013-3