Adaptive learning in complex reproducing kernel Hilbert spaces employing Wirtinger's subgradients

Area of Science:

• Signal Processing
• Machine Learning
• Complex Analysis

Background:

• Online supervised learning in complex signal processing faces challenges with non-linearities and memory requirements.
• Existing methods often lack flexibility in loss function choice and efficient handling of large datasets.

Purpose of the Study:

• To propose a generalized framework for non-linear online supervised learning with complex-valued data.
• To address memory constraints in kernel-based online learning schemes.
• To demonstrate the framework's applicability in complex signal processing tasks.

Main Methods:

• Mapping complex input data into a complex reproducing kernel Hilbert space (RKHS).
• Utilizing Wirtinger's calculus for analytical subgradient derivation of various convex loss functions.
• Employing a sparsification scheme based on projection onto closed balls to manage memory.
• Considering both linear and widely linear estimation filters within RKHS.

Main Results:

• The framework successfully handles non-linearities using complex kernels or real kernels via complexification.
• Analytical subgradients are derived for diverse loss functions, enhancing learning flexibility.
• The sparsification scheme effectively mitigates increasing memory requirements in RKHS online learning.
• Demonstrated effectiveness in non-linear channel identification, equalization, and Quadrature Phase Shift Keying (QPSK) equalization.

Conclusions:

• The proposed framework offers a robust and flexible approach to non-linear online supervised learning in complex signal processing.
• The integration of Wirtinger's calculus and sparsification provides an efficient solution for complex learning tasks.
• The framework's versatility is confirmed through successful application in various signal processing scenarios.