Treating the Filter Weights as Learnable Functions: An Efficient Nonlinear Filtering Framework and Its Adaptive Algorithms

Adaptive filters, constrained by a linear filtering framework, often struggle with nonlinear modeling in complex processes. Kernel adaptive filters (KAFs) offer a promising solution by mapping input signals into diverse feature spaces. However, their computational efficiency and filtering accuracy may still not meet the demands of practical applications. To this end, based on Kolmogorov-Arnold (KA) representation theorem, this paper proposes a novel nonlinear filtering framework by treating filter weights as learnable functions. Specifically, the learnable functions are represented as a linear combination of multiple Gaussian basis functions with different centers. To determine the coefficients that define these learnable functions, the weight-learning-based least mean square (WL-LMS) and weight-learning-based recursive least squares (WL-RLS) algorithms are further proposed based on minimum mean square error (MMSE). In addition, to ensure convergence and assess the steady-state and transient performance of proposed algorithms, a thorough theoretical analysis of the convergence conditions and excess mean square error (EMSE) is provided. Finally, linear-in-parameters system identifications validate the correctness of theoretical analysis, while chaotic time-series prediction and nonlinear system identifications demonstrate the superiorities of the proposed WL-LMS and WL-RLS algorithms.