Low complexity adaptive algorithms for Principal and Minor Component Analysis
Digital Signal Processing
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We note that the well-known Fast Rayleigh's quotient-based Adaptive Noise Subspace (FRANS), FRANS with Householder transformation (HFRANS), and fast data projection method (FDPM) algorithms all inherit from the data projection method (DPM) algorithm, but with different orthonormalization matrices. Starting from the DPM, we analyze the orthonormalization matrices of all these algorithms and develop a novel orthonormalization matrix for our algorithm. Based on this novel orthonormalization matrix, a fast and stable implementation of the DPM algorithm which has the merits of both the FRANS and FDPM approaches is investigated for principal and minor subspace tracking. The proposed algorithm can switch between the principal and minor subspace tracking with a simple sign change of its step size parameter. Moreover, it reaches the $3np$ lower bound of the dominant complexity and guarantees the orthonormality of the tracked subspace. The numerical stability of our algorithm is established theoretically and tested numerically. The strengths and weaknesses of the proposed algorithm to some existing subspace tracking algorithms are demonstrated using a de facto benchmark example. Simulation results are presented to demonstrate the effectiveness of the tracking algorithm advocated.