Algorithms for accelerated convergence of adaptive PCA.

Area of Science:

• Machine Learning
• Signal Processing
• Data Science

Background:

• Traditional principal component analysis (PCA) algorithms, often based on gradient descent, exhibit slow convergence.
• The convergence rate of existing PCA methods is sensitive to the choice of gain sequences.
• Online applications require faster PCA convergence and automated gain selection.

Purpose of the Study:

• To develop and analyze new adaptive algorithms for principal component analysis (PCA).
• To address the slow convergence and gain selection issues of traditional PCA algorithms.
• To introduce novel adaptive PCA methods derived from an unconstrained objective function.

Main Methods:

• Derivation of adaptive PCA algorithms using gradient descent, steepest descent, conjugate direction, and Newton-Raphson methods.
• Development of an unconstrained objective function for PCA.
• Global convergence proof for the adaptive gradient descent PCA algorithm using stochastic approximation theory.

Main Results:

• New adaptive algorithms for PCA were derived, including novel methods from steepest descent, conjugate direction, and Newton-Raphson approaches.
• The adaptive gradient descent PCA algorithm demonstrated faster convergence compared to traditional gradient descent methods.
• Experimental results confirmed superior convergence rates for the new algorithms on stationary and nonstationary Gaussian sequences.

Conclusions:

• The proposed adaptive PCA algorithms provide a significant improvement in convergence speed over traditional methods.
• These new algorithms are suitable for online applications requiring efficient and automatic gain selection.
• Further comparisons show the steepest descent adaptive algorithm is competitive with state-of-the-art methods.