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.

The augmented complex-valued normalized subband adaptive filtering algorithm exhibits excellent convergence performance in the processing of highly correlated non-circular complex-valued signals, but it lacks robustness against impulsive noise. Although the recently developed augmented complex-valued normalized M-estimate subband adaptive filtering (ACNMSAF) algorithm improves robustness via the complex-valued modified Huber function, its adaptive threshold and MSE equivalence under non-impulsive noise conditions cause significant performance degradation in non-Gaussian or frequent impulsive noise environments. To address this problem, this paper proposes an augmented complex-valued normalized logarithmic subband adaptive filtering (ACNLSAF) algorithm based on the relative logarithmic function. Furthermore, by equating the subband a posteriori error variance to the corresponding subband background noise variance, a subband-level variable step-size ACNLSAF (VSS-ACNLSAF) algorithm is designed, which achieves a significant improvement in convergence performance at the cost of higher computational complexity. Subsequently, the stability and steady-state excess mean square error (EMSE) of the ACNLSAF algorithm are analyzed. Extensive simulations in system identification and stereo acoustic echo cancellation (SAEC) validate the accuracy of the theoretical EMSE model and demonstrate the superiority of the proposed algorithms over the existing ACNMSAF algorithm in terms of estimation accuracy, particularly in non-Gaussian and frequent impulsive noise environments.

Conventional adaptive filter (AF) algorithms based on the stochastic gradient descent (SGD) criterion suffer severe performance degradation in the presence of impulsive noise. To tackle this issue, this letter first proposes a robust arctangent exponential (RATE) loss function to improve robustness against impulsive noise. However, AF algorithms using the RATE loss function fail to maintain stable convergence and may even diverge when the input signals are also corrupted by impulsive noise. Recently, the fractional-order SGD (FoSGD) criterion has proven effective in guaranteeing stable convergence of AF algorithms for impulsive inputs. Nevertheless, FoSGD-based AFs formulated within the Euclidean space suffer from degraded convergence rate and steady-state accuracy when processing non-Euclidean data. To overcome the above issues and speed up convergence for correlated input signals, we extend the FoSGD criterion to the Riemannian manifold optimization framework and develop the subband RATE fractional-order Riemannian SGD (SRATE-Fo-RSGD) algorithm. This algorithm degenerates into SRATE-RSGD when the fractional-order $\zeta$=1. Numerical results show that the proposed algorithms within the Fo-RSGD framework outperform conventional RSGD-based counterparts in terms of convergence and steady-state performance. To alleviate the limitation of fixed learning rate, we further propose its variable learning rate (VLR) variant, SRATE-Fo-RSGD-VLR. Finally, simulation results verify that the proposed algorithms outperform existing benchmark algorithms.

In recent years, progress in adaptive graph signal processing algorithms has provided effective solutions for processing signals defined on graph structures. As a classical strategy in information theory, the Generalized Maximum Correntropy Criterion (GMCC) exhibits good resistance to non-Gaussian noises. When non-Gaussian noise interferes with the graph signal, the graph signal processing algorithm based on GMCC (GSP GMCC) algorithm shows better performance. However, the GSP GMCC algorithm itself has three parameters that need to be manually tuned, and the process of manually tuning the parameters is complex and tedious. Meanwhile, the non-concave and non-convex nature of the GMCC function itself limits its own convergence rate and adaptive estimation accuracy. To solve the above problems, based on the strongly convex function half-quadratic criterion (HQC), the GSP HQC algorithm is proposed in this paper. The performance analysis of the GSP HQC algorithm is implemented in this paper. Simulation experiments demonstrate that the GSP HQC algorithm achieves superior performance in terms of convergence rate and adaptive estimation accuracy while maintaining computational complexity comparable to existing algorithms

In feedback active noise control (FBANC) systems, achievable noise reduction is inherently constrained by the accuracy of online secondary-path modeling (OSPM), with higher modeling precision leading to improved noise cancellation. To address this limitation, this paper proposes an Asynchronous Hierarchical Dual-Population Cooperative Optimization Algorithm (AH-DCOA) that jointly optimizes noise attenuation and secondary-path estimation accuracy. An execution-optimization asynchronous dual-layer architecture is employed to distribute computationally intensive search operations over extended evaluation frames, thereby reducing the per-sample computational burden and satisfying real-time constraints. In addition, a parameter-smoothing injection mechanism is introduced to enforce Lipschitz continuity and suppress transient disturbances caused by evolutionary updates. Experimental results demonstrate that AH-DCOA achieves effective noise reduction and faster convergence across diverse environments, ensuring stable FBANC operation without sacrificing modeling precision.

Geometric algebra (GA) constitutes an effective mathematical framework for multidimensional signal processing, as it inherently preserves geometric couplings between different signal dimensions. Existing GA-domain adaptive filtering algorithms suffer severe performance degradation under impulsive interference. To mitigate this limitation, this paper develops a GA-based minimum error entropy with fiducial points (GA-MEEF) algorithm. The proposed method generalizes the scalar MEEF cost function to multivector space by replacing scalar absolute values with multivector norms. A compact weight update rule is deduced via multivector gradient derivation, enabling joint optimization over all eight basis blades without separate blade-wise computation. Simulations under two distinct impulsive noise models demonstrate that the proposed GA-MEEF algorithm achieves a lower steady-state mean-square deviation (MSD) and a faster convergence rate than other competing algorithms, which validates the robustness superiority of the MEEF criterion under impulsive noise conditions.

Accurately estimating the state of charge (SoC) of lithium iron phosphate (LFP) batteries in a battery management system (BMS) is challenging because flat open-circuit-voltage plateaus, strong temperature effects, and ageing-induced parameter drift weaken the link between terminal measurements and internal states. Existing model-based, data-driven, and loosely coupled hybrid estimators often lose accuracy under non-stationary operating profiles, incur high computational demand, or provide poorly calibrated uncertainty, which limits their suitability for embedded BMS. Therefore, this paper proposes a deep learning-enhanced adaptive filtering framework that tightly couples a lightweight Transformer with an age-aware second-order Kalman filter. The Transformer processes operational current, voltage, temperature, and a health indicator to generate time-varying equivalent-circuit parameters, noise statistics, and bias terms, while the Kalman filter remains the sole state estimator and propagates physically interpretable uncertainty. To handle non-stationary signals within embedded hardware constraints, a de-stationary preprocessing block, grouped-query attention, and a hierarchical parameter-adaptation strategy are introduced. The framework is evaluated on a public LFP drive-cycle dataset spanning -10 degrees C to 50 degrees C, using DST and FUDS profiles for training and an unseen US06 profile for testing. On the US06 cycle, the proposed method achieves a SoC mean absolute error of 1.92% and maintains errors of about 2% across all tested temperatures, while consistently outperforming both classical filtering approaches and recent deep learning baselines. These results indicate that a tightly coupled Transformer-Kalman architecture can provide accurate, robust, and uncertainty-aware SoC estimation at a complexity compatible with real-time BMS deployment.

Widely linear (WL) adaptive filters have found widespread applications because of their excellent learning ability for non-circular signals. Recently, several effective augmented complex-valued normalized subband adaptive filtering (NSAF) algorithms have been developed to tackle with highly correlated input signals. However, these algorithms have the drawback of heavy computational burden, which is not conducive to practical application requirements. In this paper, by borrowing the online censoring strategy, we develop a novel online censoring widely linear NSAF (OC-WLNSAF) algorithm, which can significantly reduce computational overhead without causing unacceptable loss in learning performance. To guarantee the realization of the algorithm, the corresponding theoretical mean convergence range and steady-state excess mean-square error are analyzed. To eliminate the influence of impulsive noises on the learning behavior of the algorithm, a robust version, namely the robust OC-WLNSAF (ROC-WLNSAF) algorithm, is designed. Furthermore, conventional complex-valued versions of the above two algorithms including the online censoring complex-valued NSAF (OC-CNSAF) and the robust OC-CNSAF (ROC-CNSAF), are also proposed to process circular signals. Finally, numerical simulation confirms the efficiency of the presented algorithms in stereophonic acoustic echo cancellation (SAEC), wind prediction, and channel estimation applications.

Most existing nonlinear adaptive filtering algorithms only account for output noise, neglecting the fact that input noise is also prevalent in practice. Although the recently proposed bias-compensated kernel least mean square (BCKLMS) algorithm addresses input noise in the nonlinear errors-in-variables (EIV) model, it still suffers from two major limitations. First, the use of a fixed-size dictionary restricts network growth but also prevents it from fully capturing the characteristics of the input signal. Second, as an least mean square (LMS) based algorithm, it exhibits poor robustness in the presence of non-Gaussian noise in the output signal. To overcome these issues, this paper proposes the random Fourier bias-compensated filter under general adaptive function (RFFBCGA) algorithm. Within the random Fourier feature based bias-compensated (RFFBC) framework, the proposed algorithm not only maintains a fixed network structure and effectively mitigates input noise interference through the BC term, but also achieves improved characterization of the input signal. Moreover, by leveraging the flexible form of the general adaptive (GA) function, the algorithm's robustness across various noise scenarios is further enhanced. Extensive simulations, including real-world time series prediction tasks, demonstrate the superiority of the proposed method.

In In-band full-duplex (IBFD) transceivers, the motivation of using widely (augmented) linear or nonlinear structure has been elucidated by recent researches in terms of performance improvements for the digital self-interference cancellation (DSIC), a main challenge in IBFD communication systems. In this work, we first formulate a widely memory polynomial (MP)-based model by taking both power amplifier (PA) nonlinearities and inphase/quadrature (I/Q) imbalances into account, which have been generally confirmed as the most significant nonlinear imperfections in IBFD radios. In order to fast and reliably estimate the SI channel coefficients and mitigate the self interference (SI) in the digital domain, we subsequently present a MP-based proportionate widely affine projection algorithm (PWAPA) from the basis pursuit (BP) perspective. We also derive the analytical expression of the steady-state excess mean square error (EMSE), being referred as the residual SI power. In the analysis, the dependency between the filter weight vector and past noise are considered leading to more accurate results. In addition, we propose a MP-based hard thresholding PWAPA algorithm (PWAPA-Ht), which imposes a hard thresholding operator on the SI channel coefficients to select most-relevant polynomial terms in both transient and steady-state processes. The hard thresholding scheme enables the reduction of the model complexity and facilitates the online implementation. Numerical experiments based on both simulated orthogonal frequency division multiplexing (OFDM) waveforms and experimental dataset show that the proposed algorithms outperform the exiting widely nonlinear least mean square algorithm (WNLMS) and widely affine projection algorithm (WAPA) for the DSIC and verify a coincidence between theoretical results and simulations.

The ability to forecast future events in an interpretable manner is crucial for analyzing dynamic systems. Temporal knowledge graphs (TKGs) provide a structured framework for this task, where rule-based methods are prized for their transparency. However, current approaches suffer from a recency bias, relying heavily on the latest events while overlooking information about rule activations, such as their long-term frequency and short-term tendency. To address these limitations, we introduce FETA, a novel rule-based framework for explainable temporal knowledge graphs forecasting that systematically incorporates these global temporal patterns. FETA comprises two innovative components: a Frequency Enhanced Module (FEM) that refines predictions by aggregating long-term historical signals, and a Tendency Aware Module (TAM) that captures the evolving dynamics of rule utility through the divergence between long-term and short-term behaviors. Extensive experiments on benchmark datasets, including ICEWS14, ICEWS18, ICEWS05-15, and GDELT, validate that FETA achieves new state-of-the-art results, outperforming embedding-based, rule-based, and large language model-based baselines. The strong performance in cross-dataset and low-resource settings further underscores its capability for reliable and interpretable temporal forecasting.

As one of the most advanced variants in the correntropy family, the multi-kernel correntropy criterion demonstrates superior accuracy in handling non-Gaussian noise, particularly with multimodal distributions. However, current approaches suffer from key limitations-namely, reliance on a single type of sensitive Gaussian kernel and the manual selection of free parameters. To address these issues and further boost robustness, this paper introduces the concept of multi-kernel mixture correntropy (MKMC), along with its key properties. MKMC employs a flexible kernel function composed of a mixture of two Student's t-Cauchy functions with adjustable (non-zero) means. Building on this criterion within multi-sensor networks, we propose a robust distributed extended Kalman filter-AMKMMC-RDEKF based on adaptive multi-kernel mixture maximum correntropy. To reduce communication overhead, a consensus averaging strategy is incorporated. Furthermore, an adaptive mechanism is introduced to mitigate the impact of manually tuned free parameters. At the same time, the computational complexity and convergence ability of the proposed algorithm are analyzed. The effectiveness of the proposed algorithm is validated through challenging scenarios involving power system and land vehicle state estimation.

Adaptive filters have been successfully applied in uniform linear array system to estimate the direction-of-arrival. However, traditional adaptive filtering algorithms suffer from degraded estimation accuracy against impulsive noise interferences or require a trade-off between the convergence speed and estimation accuracy. This article proposes a variable step-size general bias-compensated M-estimate (VSS-GBCM) adaptive filtering algorithm to address the above issues. This article first derives a bias-compensated M-estimate (BCM) adaptive filtering algorithm from a continuous bias-compensated M-estimate cost function, and then analyze its transient and steady-state mean square performance under common statistical assumptions. Sub-sequently, a variable step-size (VSS) mechanism is obtained by minimizing the mean square deviation at every snapshot, result ing in a VSS-BCM adaptive filtering algorithm. Moreover, the VSS-GBCM is developed by solving a general bias-compensated M-estimate cost function and using the above VSS strategy, where the VSS-BCM is its special case. Finally, simulation results validate that our solutions are superior to competing methods.

Adaptive filter in complex scenarios demands algorithms that integrate fast convergence, low complexity, and robust performance under diverse noise conditions. To address this challenge, we propose an online censoring robust total generalized adaptive filter using improved data-reused method (RTGAIDROC) algorithm. The proposed RTGA variant possesses the advantages of both the total least squares (TLS) strategy and the robust generalized adaptive (RGA) function. This algorithm not only effectively handles input noise under the errors-invariables (EIV) model but also achieves excellent performance across diverse noise environments. Furthermore, to meet the high demand for convergence speed in practical applications, an improved data reuse (IDR) method is introduced, enabling faster convergence in the early stages of iteration without compromising steady-state performance. The increased computational complexity brought by the IDR method is mitigated using the online censoring (OC) strategy. We also modify the OC threshold for real-valued algorithms, as the original threshold was defined for the complex domain. Beyond these algorithmic enhancements, a local stability analysis for the proposed algorithm is provided, and the theoretical steady-state mean-square deviation (MSD) is derived. Finally, simulation experiments in system identification and acoustic echo cancellation (AEC) scenarios validate the superior performance of the proposed algorithm.

The distributed Kalman filter (DKF) has shown outstanding performance in Gaussian noise environments, making it a suitable choice for handling Gaussian noise in state estimation problems through information fusion among nodes in sensor network. Nevertheless, when the measured value is contaminated by an outlier in nonlinear systems, it causes a situation where non-Gaussian noise is present, which will lead to degradation of the conventional DKF's performance. Therefore, to solve the problem of outlier in nonlinear systems, a centralized outlier-robust extended Kalman filter (C-OR-EKF) is developed, which adopts the M-estimation cost function as the optimal criterion to deal with the error in the filtering process. The M-estimation effectively removes the effect of large outliers, thus making the algorithm robust against non-Gaussian noise. Furthermore, to reduce the node communication burden, a distributed outlier-robust extended Kalman filter (D-OR-EKF) is proposed. Moreover, the mean error and mean square error behavior of C-OR-EKF and D-OR-EKF are analyzed separately to determine the stability of the proposed methods. Finally, by conducting experiments in a nonlinear system and an IEEE 14 system, the D-OR-EKF based on different cost functions of M-estimation is compared with the existing distributed robust Kalman filter. The experimental results show that the proposed algorithms effectively handle the problem of state estimation in non-Gaussian noise environment.

Currently, there is relatively limited research on algorithms that can maintain excellent performance when dealing with correlated input signals and output disturbances caused by generalized Gaussian noise. To further enhance the steady-state performance of the existing generalized maximum correntropy criterion normalized subband adaptive filter (GMCC-NSAF) algorithm, this paper proposes the generalized modified Blake-Zisserman normalized subband adaptive filter (GMBZ-NSAF) algorithm based on the GMBZ cost function. The proposed algorithm not only effectively addresses the issue of input signal correlation but also achieves superior performance compared to GMCC-NSAF when the output is corrupted by generalized Gaussian noise. In addition, a detailed theoretical analysis of the mean and mean-square performance of the GMBZ-NSAF algorithm is provided. Finally, simulation results under various noise conditions demonstrate the outstanding performance of the proposed algorithm.

Recently, the nearest Kronecker product (NKP) decomposition-based normalized least mean square (NLMS NKP) algorithm has demonstrated superior convergence performance compared to the conventional NLMS algorithm. However, its convergence rate exhibits significant degradation when processing highly correlated input signals. To address this problem, we propose a type-I NKP-based normalized subband adaptive filter (NSAF) algorithm, namely NSAF-NKP-I. Nevertheless, this algorithm incurs substantially higher computational overhead than the NLMS-NKP algorithm. Remarkably, our enhanced type-II NKP-based NSAF (NSAF-NKP-II) algorithm achieves equivalent convergence performance while substantially reducing computational complexity. Furthermore, to enhance robustness against impulsive noise interference, we develop two robustvari ants: the maximum correntropy criterion-based robust NSAF NKP (RNSAF-NKP-MCC) and logarithmic criterion-based robust NSAF-NKP (RNSAF-NKP-LC) algorithms. Additionally, detailed analyses of computational complexity, step-size range, and theoretical steady-state performance are provided for the proposed algorithms. To enhance the practicability of the NSAF NKP-II algorithm in complex nonlinear environments, we further devise two nonlinear implementations: the trigonometric functional link network-based NKP-NSAF (TFLN-NSAF-NKP) and Volterra series expansion-based NKP-NSAF (Volterra-NKP NSAF) algorithms. In active noise control (ANC) systems, we further propose the filtered-x NSAF-NKP-II (NKP-FxNSAF) algorithm. Simulation experiments in echo cancellation, sparse system identification, nonlinear processing, and ANC scenarios are conducted to validate the superiority of the proposed algorithms over existing state-of-the-art counterparts.

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 cubature Kalman filter based on minimum error entropy (MEE-CKF) offers accurate and robust performance in state of charge (SOC) estimation. However, due to the inflexibility of the minimum error entropy (MEE), this algorithm demonstrates limited robustness when confronted with more complex noise environments. To address these limitations, this paper proposes a generalized mixture minimum error entropy-based (GMMEE) square-root cubature Kalman filter (GMMEE-SRCKF). The square-root algorithm ensures improved numerical stability and avoids covariance degeneration, while the GMMEE criterion with two flexible kernels adapts effectively to non-Gaussian noise. Moreover, a hybrid tree seed and genetic algorithm (TSGA) is introduced to optimize the kernel parameters automatically. Experimental results confirm that the TSGA-optimized GMMEE-SRCKF outperforms existing robust filters, achieving the root mean square error (RMSE) of less than 0.5