---
title: Least Mean p-Power Algorithm
url: https://www.emergentmind.com/topics/least-mean-p-power-algorithm
type: topic
---

# Least Mean p-Power Algorithm

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 $0 < p \leq 2$ [1307.7226][2405.04111][1503.00792].

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 [1307.7226][2405.04111][1503.00792].

## 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 [1503.00792]. 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 [1503.00792].

## 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 [1307.7226]. 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 [1608.02060].

## 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 $v(n) = |\varepsilon(n, n)|^{p-2}$ [2202.00560].

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 [2202.00560].

## 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 $t$ [2405.04111]. 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 [2405.04111].

## 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 [1603.09022].

## 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 [1307.7226][2202.00560].
- 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 $p$ [1503.00792][2405.04111].
- In distributed/weighted settings, additional parameters for weight updates require careful tuning to balance adaptation in heterogeneous noise fields [1608.02060].
- 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 [2202.00560][1503.00792][1307.7226][2405.04111].

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.

Source: https://www.emergentmind.com/topics/least-mean-p-power-algorithm