Independent Component Analysis Algorithms Based on Maximum Likelihood Estimation

In this article, we provide a comprehensive exploration of independent component analysis algorithms based on maximum likelihood estimation. These algorithms utilize the statistical properties among multiple random variables to decompose them into independent components. We examine three distinct implementations: the stochastic gradient algorithm, which incrementally updates parameters using random data subsets for memory efficiency; the relative gradient algorithm, which employs natural gradient descent for improved convergence properties; and the fast fixed-point algorithm (FastICA), known for its cubic convergence rate and stability through approximate Newton iterations.

Each algorithm's implementation methodology is detailed, including key computational steps such as whitening preprocessing, non-linearity selection for density estimation, and convergence criteria. The stochastic gradient approach typically involves iterative weight updates using a learning rate schedule, while the relative gradient method computes updates based on the Riemannian geometry of the parameter space. FastICA implementation focuses on orthogonalization techniques and efficient eigenvalue decomposition for separation matrix estimation.

We analyze the advantages and limitations of each algorithm, including computational complexity, convergence speed, and sensitivity to initial conditions. Selection criteria for different application scenarios are discussed, considering factors like data dimensionality, real-time processing requirements, and noise characteristics. This article aims to deepen understanding of ICA methodologies and provide practical guidance for research and implementation projects.