ICA Algorithm: Implementation with Maximum Likelihood and BFGS Optimization

In this document, we introduce the ICA algorithm. This algorithm implements Bell and Sejnowski's 1995 Infomax approach using a maximum likelihood formulation. The implementation assumes no noise presence and requires that the number of observations matches the number of sources. For optimization, we employ the BFGS quasi-Newton method [2], which efficiently handles the likelihood maximization through gradient-based updates. When computing independent components, we utilize the Bayes Information Criterion (BIC) [3] to determine the optimal number of independent components, while Principal Component Analysis (PCA) is applied for dimensionality reduction prior to ICA processing. The typical implementation workflow involves: first reducing data dimensions using PCA's eigenvalue decomposition, then applying ICA optimization with BFGS to maximize statistical independence between components. Notably, the ICA algorithm serves as a powerful tool for blind source separation of mixed signals. By decomposing mixed signals into statistically independent components, we can better understand signal characteristics and perform more precise analysis. In practical applications, ICA finds extensive use in image processing, signal processing, speech recognition, and other domains where source separation is crucial. Therefore, understanding ICA's principles and applications is essential, particularly for professionals aiming to succeed in related technical fields.