An Adaptive Self-Stabilizing Algorithm for Minor Generalized Eigenvector Extraction and Its Convergence Analysis

Abstract

Generalized eigendecomposition, which extracts the generalized eigenvector from a matrix pencil, is a powerful tool and has been widely used in many fields, such as data classification and blind source separation. First, to extract the minor generalized eigenvector (MGE), we propose a deterministic discrete-time (DDT) system. Unlike some existing systems, the proposed DDT system does not need to normalize the weight vector in each iteration, since the weight vectors in the proposed DDT system are self-stabilizing. Second, we propose an adaptive algorithm corresponding to the proposed DDT system. Moreover, we study the dynamic behavior and convergence properties of the proposed DDT system and prove that the weight vector must converge to the direction of the MGE of a matrix pencil under some mild conditions. Numerical simulations show that the proposed algorithm has a better performance in terms of convergence speed and estimation accuracy than some existing algorithms. Finally, we conduct two experiments on real data sets to demonstrate its practicability.

Novel Dual-Purpose Algorithm for Principal and Minor Component Analysis

“…Remark: The DDT system of the proposed algorithm has a computation complexity of n 2 + 4n flops per update, which is the same as n 2 + 4n of the Chen algorithm's DDT system [14], and is cheaper than n 2 + 8n of the Hasan algorithm's DDT system in [15] and n 2 +5n the projection approximation subspace tracking with deflation (PASTd) algorithm's DDT system in [16]. In addition, the operations involved in (4) are simple matrix addition and multiplication, which are easy for the systolic array implementation.…”

“…Dynamic property analysis, which can describe the movement trajectory of the weight vector of a neural network during the whole iteration procedure and can help us understand why algorithms can converge to the desired component after iterations, has become an important aspect of studying neural network algorithms. In recent years, the deterministic discrete time (DDT) analysis method has become the mainstream method for analyzing the dynamic properties of algorithms [13], [16], [17]. When the DDT method is employed, the dynamic property of a weight vector can be obtained by projecting the weight vector onto all of the eigenvectors of the autocorrelation matrix and analyzing the changing laws of these projections.…”

Online Dominant Generalized Eigenvectors Extraction Via A Randomized Method

“…The generalized Hermitian eigenvalue problem (GHEP) [1] is of great interest in signal processing, machine learning and data analysis applications. The GHEP algorithms provide powerful tools to treat problems in blind source separation [2,3], feature extraction [4,5], noise filtering [6], fault detection [7], antenna array processing [8], classification [9], and speech enhancement [10]. Traditional methods for solving the GHEP include power and inverse iteration based methods, Lanczos method and Jacobi-Davidson method [1,11].…”