Least Mean p-Power Algorithm

• The Least Mean p-Power algorithm is an adaptive filtering method that minimizes the p-th power of the error to enhance robustness against impulsive, heavy-tailed noise.
• It extends the traditional LMS algorithm with normalized, sparsity-aware, and distributed frameworks to improve stability and convergence in diverse noise conditions.
• Advanced variations like recursive, logarithmic, and graph-based neural adaptations illustrate its practical significance in addressing real-world non-Gaussian interference challenges.

The Least Mean $p$-Power (LMP) algorithm is a broad class of adaptive filtering algorithms designed to enhance robustness against impulsive and non-Gaussian noise by using a $p$-norm-based error criterion. Generalizing the standard Least Mean Squares (LMS) algorithm, LMP methods are characterized by the minimization of the $p$-th power of the instantaneous error, with $0 < p \leq 2$, where $p < 2$ is typically selected for heavy-tailed or $\alpha$-stable noise environments. The LMP family supports various extensions, including normalized variants, sparsity-aware penalties, variable power strategies, and distributed/diffusion frameworks, making it foundational in robust adaptive signal processing and control.

1. Fundamental Principle and Algorithmic Structure

The LMP algorithm minimizes the expected $p$-th power of the instantaneous error: $J(w) = \mathbb{E}[|d(n) - w^T x(n)|^p]$ where $d(n)$ is the observed scalar, $x(n)$ the input or regressor vector, $w$ the coefficient vector, and $0 < p \leq 2$ (Wen, 2013, Yan et al., 2024, Ma et al., 2015).

The stochastic gradient update is: $w(n+1) = w(n) + \mu\,|e(n)|^{p-2}\,e(n)\,x(n)$ with instantaneous error $e(n) = d(n) - w(n)^T x(n)$ and step-size $\mu$. For $p=2$, this reduces to the conventional LMS algorithm. As $p \to 1$, the update increasingly discounts large error outliers, enhancing robustness to impulsive noise but reducing convergence speed in Gaussian noise (Wen, 2013, Yan et al., 2024, Ma et al., 2015).

2. Normalized and Sparsity-Aware Variants

Normalized LMP (NLMP) adapts to variations in input signal power by normalizing the update: $$w(n+1) = w(n) + \mu\,\frac{e(n)|e(n)|^{p-2} x(n)}{\epsilon + \|x(n)\|_2^p}$$ where $\epsilon>0$ prevents numerical instability (Ma et al., 2015). To further exploit systems with sparse unknown filters, sparsity-aware extensions such as CIMNLMP introduce regularizers that approximate the $\ell_0$ norm via correntropy-induced metrics: $$P_{\text{CIM}}(w) = \sum_{i=1}^L \left(1 - \exp(-w_i^2 / (2\sigma^2))\right)$$ with update: $$w(n+1) = w(n) + \text{NLMP-update} - \lambda\,\nabla P_{\text{CIM}}(w)$$ where $\lambda$ modulates the sparsity penalty and $\sigma$ is the CIM kernel width. Adaptive regularization (CIMVRNLMP) uses a variable $\epsilon(n)$ dependent on recent error energy, further widening stability and convergence regimes (Ma et al., 2015).

3. Diffusion, Distributed, and Weighted Extensions

Diffusion LMP algorithms extend the single-node LMP principle to distributed networks. Nodes collaboratively estimate a shared parameter vector using local gradients and information from neighbors. Principal strategies include Adapt-Then-Combine (ATC) and Combine-Then-Adapt (CTA), employing local combination weights for robustness: $$\psi_{k,n} = w_{k,n-1} + \mu_k \sum_{l\in\mathcal{N}_k} c_{lk}|e_{l,n}|^{p-2}e_{l,n}u_{l,n}$$

$$w_{k,n} = \sum_{l\in\mathcal{N}_k} a_{lk}\psi_{l,n}$$

where $w_{k,n}$ and $\psi_{k,n}$ are, respectively, the adapted and intermediate estimates at node $k$; $a_{lk}$ and $c_{lk}$ are convex combination weights (Wen, 2013). Weighted diffusion LMP introduces adaptive node-specific weights $\alpha_k(n)$, learned via steepest-descent, to emphasize reliable nodes in non-uniform or heteroscedastic noise (Zayyani et al., 2016).

4. Recursive Least $p$-Power and Logarithmic Generalizations

The filtered-x recursive least $p$-power (FxRLP) algorithm addresses active noise control under impulsive interference. FxRLP minimizes an exponentially weighted sum of $p$-power a posteriori errors: $$J_{\text{FxRLP}}(n) = \sum_{i=1}^n \lambda^{n-i} |\varepsilon(i, n)|^p$$ with filtered reference vector $x_s(i)$. Recursive normal equations employ the gain vector

$$K(n) = \frac{v(n) P(n-1) x_s(n)}{\lambda + v(n)x_s^T(n)P(n-1)x_s(n)}$$

where $v(n) = |\varepsilon(n, n)|^{p-2}$ (Zheng et al., 2022).

To further attenuate outlier influence, the filtered-x logarithmic recursive least $p$-power (FxlogRLP) employs

$$J_{\text{FxlogRLP}}(n) = \sum_{i=1}^n \lambda^{n-i} [\log(1 + |\varepsilon(i, n)|)]^p$$

with a modified weighting factor $$v(n) = [\log(1+|\varepsilon(n, n)|)]^{p-1} / [(1+|\varepsilon(n, n)|)\,|\varepsilon(n, n)|]$$. FxlogRLP has demonstrated superior convergence rate and higher steady-state noise reduction (measured via averaged noise reduction ANR) compared to FxLMP, FxRLS, and FxRLP in impulsive noise environments (Zheng et al., 2022).

5. Graph Signal Processing and LMP-based Neural Frameworks

The adaptive Least Mean $p$th Power Graph Neural Network (LMP-GNN) framework generalizes LMP updates to graph-structured signal estimation problems. The learning objective is the instantaneous $\ell_{p}$-norm of error restricted to observed nodes: $J = \|\epsilon[t]\|_p^p = \sum_i |\epsilon_i[t]|^p$ where $\epsilon[t]$ denotes the error on observed graph nodes at time $t$ (Yan et al., 2024). The LMP-GNN architecture integrates graph convolutions in both forward inference and online parameter adaptation, with parameter matrices $\Theta_{f,l}$ governing the spectral filters. Sign-GNN, corresponding to $p=1$, emphasizes maximal robustness to outliers in online graph estimation. LMP-GNN outperforms conventional adaptive GSP algorithms and standard GCN/STGCN models on tasks involving impulsive, heavy-tailed, and missing data (Yan et al., 2024).

6. Variable-$p$ and Adaptive Power Strategies

In sparse system identification, the variable-$p$ norm LMS algorithm dynamically adjusts the exponent $p$ to optimize the balance between convergence and steady-state misadjustment: $J(w) = \frac{1}{2} e(n)^2 + \rho_p \|w\|_p^p$ with $p$ iteratively adapted according to the gradient of the root relative deviation (RRD) of the tap estimates: $$p(n+1) = p(n) + \delta_n [\text{RRD}_n - \text{RRD}_{n-1}]$$ where the step-size $\delta_n$ is annealed to ensure stability. This method automatically modulates the effective penalty as system sparsity changes, achieving lower steady-state error without sacrificing convergence speed in a wide range of sparsity and SNR conditions (Feng et al., 2016).

7. Parameter Selection, Stability, and Performance Benchmarks

The power exponent $p$ is critical for algorithm robustness:

• For $\alpha$-stable noise with exponent $\alpha$, $p\lesssim\alpha$ ensures finite $p$th moments and optimal robustness (Wen, 2013, Zheng et al., 2022).
• Step-size $\mu$ and normalization parameter $\epsilon$ must be chosen to satisfy convergence constraints, analogous to those for LMS but with additional consideration for impulsive environments and the choice of $p$ (Ma et al., 2015, Yan et al., 2024).
• In distributed/weighted settings, additional parameters for weight updates require careful tuning to balance adaptation in heterogeneous noise fields (Zayyani et al., 2016).
• Empirical evaluations consistently indicate that LMP and its variants surpass LMS and RLS-type algorithms in environments with impulsive, non-Gaussian, or non-uniform noise, achieving faster convergence and lower steady-state error provided that $p$ is appropriately selected (Zheng et al., 2022, Ma et al., 2015, Wen, 2013, Yan et al., 2024).

These results collectively establish the LMP algorithm and its recursive, distributed, neural, and sparsity-aware extensions as the methods of choice for many robust adaptive signal processing tasks subject to impulsive and heavy-tailed disturbances.