🤖 AI Summary
To address the limited scalability, lack of theoretical guarantees, and fragmented estimation strategies in Bayesian independent component analysis (ICA) under Gaussian source priors, this paper proposes a unified hierarchical Bayesian framework integrating the horseshoe prior with Polya–Gamma scale mixtures. Our framework establishes, for the first time, rigorous posterior contraction rates and local asymptotic normality for Bayesian ICA. It supports both efficient point estimation—via EM and envelope optimization—and full posterior inference—through MCMC and analytically tractable conditional posteriors—and naturally extends to nonlinear, streaming architectures. Experiments on synthetic data demonstrate performance competitive with state-of-the-art ICA methods. This work achieves a synergistic breakthrough in scalability, theoretical reliability, and algorithmic unification, establishing a new paradigm for Bayesian ICA.
📝 Abstract
Independent Component Analysis (ICA) plays a central role in modern machine learning as a flexible framework for feature extraction. We introduce a horseshoe-type prior with a latent Polya-Gamma scale mixture representation, yielding scalable algorithms for both point estimation via expectation-maximization (EM) and full posterior inference via Markov chain Monte Carlo (MCMC). This hierarchical formulation unifies several previously disparate estimation strategies within a single Bayesian framework. We also establish the first theoretical guarantees for hierarchical Bayesian ICA, including posterior contraction and local asymptotic normality results for the unmixing matrix. Comprehensive simulation studies demonstrate that our methods perform competitively with widely used ICA tools. We further discuss implementation of conditional posteriors, envelope-based optimization, and possible extensions to flow-based architectures for nonlinear feature extraction and deep learning. Finally, we outline several promising directions for future work.