Robust Independent Component Analysis by Iterative Maximization of the Kurtosis Contrast with Algebraic Optimal Step Size (1002.3684v1)

Abstract: Independent component analysis (ICA) aims at decomposing an observed random vector into statistically independent variables. Deflation-based implementations, such as the popular one-unit FastICA algorithm and its variants, extract the independent components one after another. A novel method for deflationary ICA, referred to as RobustICA, is put forward in this paper. This simple technique consists of performing exact line search optimization of the kurtosis contrast function. The step size leading to the global maximum of the contrast along the search direction is found among the roots of a fourth-degree polynomial. This polynomial rooting can be performed algebraically, and thus at low cost, at each iteration. Among other practical benefits, RobustICA can avoid prewhitening and deals with real- and complex-valued mixtures of possibly noncircular sources alike. The absence of prewhitening improves asymptotic performance. The algorithm is robust to local extrema and shows a very high convergence speed in terms of the computational cost required to reach a given source extraction quality, particularly for short data records. These features are demonstrated by a comparative numerical analysis on synthetic data. RobustICA's capabilities in processing real-world data involving noncircular complex strongly super-Gaussian sources are illustrated by the biomedical problem of atrial activity (AA) extraction in atrial fibrillation (AF) electrocardiograms (ECGs), where it outperforms an alternative ICA-based technique.

• The paper introduces RobustICA, a novel ICA method based on the iterative maximization of the kurtosis contrast function using an algebraic optimal step size.
• It enhances computational efficiency by precisely determining the optimal step via solving a fourth-degree polynomial, eliminating prewhitening and reducing biases.
• Empirical results demonstrate its superior performance in fast convergence and effective extraction in real and complex signal scenarios, including biomedical applications.

Overview of Robust Independent Component Analysis

This paper presents a method for performing Independent Component Analysis (ICA) through an algorithm named RobustICA. The authors introduce an innovative approach to deflationary ICA, characterized by the iterative maximization of the kurtosis contrast function with an algebraically optimal step size. Deflationary ICA strategies, such as the popular one-unit FastICA algorithm, traditionally extract independent components sequentially. RobustICA diverges from these methods by applying a precise line search optimization of the kurtosis contrast function. The step size corresponding to the global maximum of the contrast along the search direction is determined among the roots of a fourth-degree polynomial, enabling a computationally efficient global optimization process.

Key Contributions

1. Optimization Technique: RobustICA employs an exact line search strategy to optimize kurtosis. The approach allows the computation of the optimal step size analytically as the solution to a quartic polynomial, significantly enhancing computational efficiency compared to iterative numerical optimizations.
2. Prewhitening Avoidance: This method eliminates the need for prewhitening, a preprocessing step that can introduce asymptotic performance limitations and biases due to residual source correlations.
3. Application to Various Signal Types: RobustICA is versatile, providing effectiveness in both real and complex-valued signal mixes, catering to non-circular and super-Gaussian sources without requiring algorithmic modifications.
4. Convergence and Cost Efficiency: The technique demonstrates robustness against local extremum entrapment and exhibits fast convergence speed particularly for short data records. This efficiency is attributed to its algebraic approach, delivering high source extraction quality with reduced computational operations per iteration compared to other ICA methods.
5. Empirical Validation: The paper substantiates the method’s performance through comprehensive numerical simulations, illustrating superior robustness and efficiency over FastICA. Additionally, it showcases a biomedical application in electrocardiography, specifically in the extraction of atrial activity from ECG recordings during atrial fibrillation (AF) episodes, demonstrating significant practical applicability.

Implications and Future Work

The development of RobustICA addresses several limitations associated with traditional ICA methods, especially under scenarios of limited data or complex signal types. By bypassing prewhitening, the algorithm presents scalability to underdetermined or convolutive mixtures, paving the way for broader applicability across disciplines from communications to biomedical engineering.

From a theoretical perspective, the introduction of algebraic step-size optimization as a practical tool for ICA extends the methodological repertoire available for efficient blind source separation. Practically, RobustICA's empirical success in AF signals extraction points towards potential advancements in clinical diagnostics and monitoring.

Future research could aim at extending RobustICA's principles to convolutive mixtures and investigating robust cumulant estimation techniques to enhance outlier resilience. These enhancements could further amplify the technique’s utility in diverse real-world applications, contributing to robust signal decomposition solutions in increasingly complex environments.