Generalized Correntropy for Robust Adaptive Filtering (1504.02931v1)

Published 12 Apr 2015 in stat.ML, cs.IT, and math.IT

Abstract: As a robust nonlinear similarity measure in kernel space, correntropy has received increasing attention in domains of machine learning and signal processing. In particular, the maximum correntropy criterion (MCC) has recently been successfully applied in robust regression and filtering. The default kernel function in correntropy is the Gaussian kernel, which is, of course, not always the best choice. In this work, we propose a generalized correntropy that adopts the generalized Gaussian density (GGD) function as the kernel (not necessarily a Mercer kernel), and present some important properties. We further propose the generalized maximum correntropy criterion (GMCC), and apply it to adaptive filtering. An adaptive algorithm, called the GMCC algorithm, is derived, and the mean square convergence performance is studied. We show that the proposed algorithm is very stable and can achieve zero probability of divergence (POD). Simulation results confirm the theoretical expectations and demonstrate the desirable performance of the new algorithm.

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Summary

The paper proposes a generalized correntropy measure using the generalized Gaussian density to robustly handle non-Gaussian noise.
The paper develops the GMCC algorithm, which achieves zero probability of divergence and superior stability compared to traditional methods.
The paper validates its approach with theoretical analysis and simulations, demonstrating lower steady-state EMSE and improved performance in challenging signal environments.

Generalized Correntropy for Robust Adaptive Filtering: An Expert Overview

The paper "Generalized Correntropy for Robust Adaptive Filtering" by Badong Chen et al. presents an advancement in adaptive filtering through the introduction of a generalized form of correntropy. The researchers explore the application of the generalized maximum correntropy criterion (GMCC) using the generalized Gaussian density (GGD) function as the kernel. This initiative aims to enhance the performance of adaptive filtering in non-Gaussian environments.

Background

Adaptive filtering is pivotal in signal processing and machine learning applications, where selecting an effective cost function is critical. Traditional measures like the mean square error (MSE) assume Gaussian distributions, often leading to suboptimal performance in the presence of non-Gaussian data. Non-Gaussian distributions can be categorized into light-tailed and heavy-tailed distributions, requiring alternative approaches for robust performance.

Key Contributions

Generalized Correntropy: The authors propose using the generalized Gaussian density function rather than the traditional Gaussian kernel. This approach broadens the application of correntropy, allowing it to adapt to various signal distribution characteristics by modifying the shape parameter $\alpha$ .
Generalized Maximum Correntropy Criterion (GMCC): The GMCC is introduced as a new cost function for adaptive filtering. The GMCC filters enhance stability and performance by aiming for zero probability of divergence (POD) and robust handling of outliers.
Algorithm Development: The paper derives the GMCC algorithm and investigates its mean-square convergence. It reveals that the GMCC can achieve a zero POD, even in challenging signal environments, due to its insensitivity to large outliers.
Theoretical and Empirical Validation: A theoretical analysis of the GMCC algorithm is provided, including a derived expression for the steady-state excess mean square error (EMSE). Simulations confirm the superior performance and stability of GMCC over traditional adaptive algorithms.

Numerical Results and Claims

The paper presents robust numerical results showcasing the GMCC algorithm's capability to outperform traditional LMS and LMF algorithms in various noise environments. The GMCC algorithm shows promising features, such as adaptability to different noise types and enhanced stability metrics, achieving convergence where others may diverge.

Implications and Future Directions

The generalized form of correntropy has potential applications beyond adaptive filtering. Its flexibility in handling various non-Gaussian noise distributions can significantly impact technologies in communications, biomedical signal processing, and other fields requiring robust estimation methods. Future developments could focus on exploring the nuances of different kernel functions beyond the GGD to further customize and optimize performance for specific applications.

Further investigation into the GMCC framework could also extend to non-linear adaptive filtering environments and the integration of deep learning frameworks, potentially providing a method for real-time adaptation within these complex systems. The development of computational efficiency techniques will address the additional computational demands posed by GMCC in practical implementations.

In conclusion, the paper provides a substantial contribution to the adaptive filtering field, offering a versatile and robust criterion for enhancing filtering performance in non-Gaussian and impulsive noise environments.

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