In this letter, we propose a hyperparameter optimization method for adaptive filtering based on deep unrolling, termed the deep unrolling affine projection (DAP) algorithm. The core idea is to reformulate the iterative structure of the traditional affine projection (AP) algorithm as a multilayer neural network, where each layer corresponds to one iteration and the step size is treated as a trainable parameter. These parameters are optimized through end-to-end supervised learning to enhance convergence speed and steady-state performance. While maintaining the interpretability and computational structure of the original algorithm, DAP leverages modern deep learning techniques to automatically learn hyperparameters from training data. Simulation results for the system identification task demonstrate that DAP outperforms the conventional AP algorithm in both convergence rate and accuracy. Importantly, DAP introduces no additional computational burden since the test phase involves only forward propagation. This makes it an efficient and practical solution for real-time adaptive filtering in engineering applications.

Traditional single-kernel or fixed-center multi kernel collaborative correntropies fundamentally assume that errors primarily cluster around a central point (typically zero). However, in real-world complex noise environments—such as those generated by mixed interference sources with diverse mechanisms—errors may exhibit multi-modal or highly asymmetric statistical characteristics. In such cases, a single central point or multi-kernels fixed at the origin cannot effectively capture the true shape of the error distribution. To address these problems, this letter proposes a novel robust learning algorithm by introducing variable-center multi-kernel correntropy into an asymmetric correntropy framework, where the kernel centers can be positioned at arbitrary locations. Compared with the maximum asymmetric correntropy criterion (MACC) algorithm, the proposed approach offers a more generalized formulation that enhances its capability to handle more complex error distributions, thereby improving algorithm performance. Notably, existing literature has not yet provided theoretical analysis for such variable-center multi-kernel asymmetric correntropy robust algorithms. Therefore, the main contributions of this work include: conducting the first theoretical analysis of the proposed algorithm, and validating the effectiveness of the analytical methodology.

The adaptive filtering algorithm based on the maximum correntropy criterion (MCC) is very effective in suppressing non-Gaussian noises and therefore attracts widespread attentions. At present, some works have been done for study the convergence and steady-state performance analysis of the MCC algorithm, but its transient performance analysis is still an open problem. To provide a comprehensive theoretical foundation for the MCC algorithm, we propose a method for transient performance analysis based on moment generating function (MGF). Since this method can efficiently calculate the expected value of the exponential term in the iterative update equation, it can avoid the discrepancies caused by introducing some approximation methods such as Taylor expansions in the analysis process. To date, there is no precedent for using this method to analyze the transient performance of the MCC algorithm. In addition, the steady-state performance and stability conditions of the MCC algorithm are discussed based on this method. Finally, the proposed analytical method is applied to the system identification problem, and the results show that the theoretical analysis results are agree well with the Monte Carlo simulation results.