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

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...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2013-03, Vol.61 (6), p.1404-1418
Main Authors: Tuan Duong Nguyen, Yamada, I.
Format: Article
Language:English
Subjects:
Adaptive algorithm
Adaptive algorithms
Algorithm design and analysis
Algorithms
Applied sciences
Approximation algorithms
Convergence
DDT
Detection, estimation, filtering, equalization, prediction
deterministic discrete-time (DDT) approach
Effectiveness
Eigenvalues
Eigenvalues and eigenfunctions
Exact sciences and technology
generalized Hermitian eigenvalue problem (GHEP)
Information, signal and communications theory
Mathematical models
minor generalized eigenvector
Newton's method
principal generalized eigenvector
Signal and communications theory
Signal processing
Signal processing algorithms
Signal, noise
Studies
Telecommunications and information theory
Transaction processing
Vectors
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Summary: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.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2012.2234744