Beyond independent component analysis: identifiability and algorithms

2025-10-08

AI Summary

Traditional independent component analysis (ICA) relies heavily on the strong global statistical independence assumption, limiting its applicability to real-world scenarios where sources exhibit only weaker dependence structures. Method: This paper proposes *pairwise mean independence* as the weakest feasible identifiability condition for blind source separation (BSS), and develops an algebraically tractable framework based on the zero-structure properties of cumulant tensors. We formulate a least-squares optimization problem over the orthogonal group to recover latent variables. Contribution/Results: We prove that pairwise mean independence constitutes the maximal relaxation boundary for identifiability—encompassing classical ICA as a special case while rigorously excluding strictly weaker assumptions that lead to non-identifiability. Experiments demonstrate that the proposed method achieves significantly improved stability and robustness in recovering non-independent source signals, thereby substantially relaxing the stringent independence constraints inherent in classical ICA.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Independent Component Analysis (ICA) is a classical method for recovering latent variables with useful identifiability properties. For independent variables, cumulant tensors are diagonal; relaxing independence yields tensors whose zero structure generalizes diagonality. These models have been the subject of recent work in non-independent component analysis. We show that pairwise mean independence answers the question of how much one can relax independence: it is identifiable, any weaker notion is non-identifiable, and it contains the models previously studied as special cases. Our results apply to distributions with the required zero pattern at any cumulant tensor. We propose an algebraic recovery algorithm based on least-squares optimization over the orthogonal group. Simulations highlight robustness: enforcing full independence can harm estimation, while pairwise mean independence enables more stable recovery. These findings extend the classical ICA framework and provide a rigorous basis for blind source separation beyond independence.

Problem

• Identifies pairwise mean independence as the minimal identifiable relaxation of independence
• Proposes algebraic algorithm for blind source separation beyond classical ICA
• Demonstrates robustness advantages over enforcing full independence assumptions

Innovation

• Pairwise mean independence enables identifiable latent recovery
• Algebraic algorithm uses orthogonal group least-squares optimization
• Relaxed independence improves robustness over classical ICA