---
title: Set-Membership Affine Projection Algorithm
url: https://www.emergentmind.com/topics/set-membership-affine-projection-algorithm
type: topic
---

# Set-Membership Affine Projection Algorithm

Set-Membership Affine Projection Algorithm (SM-AP) is a set-membership adaptive filtering method that combines affine projection with bounded-error feasibility constraints. Rather than updating the coefficient vector at every iteration, it updates only when the current estimation error violates a prescribed bound, and the active update is obtained as the minimum-disturbance correction that satisfies posterior error constraints over a block of recent data. In the literature, SM-AP is positioned between single-vector set-membership NLMS-type methods and RLS-type set-membership schemes: it reuses multiple recent regressors to accelerate convergence while retaining the data-selective update mechanism characteristic of set-membership filtering [2005.10124][1301.0097].

## 1. Set-membership formulation and the role of affine projection

The set-membership viewpoint is organized around feasibility sets induced by an error bound. In the standard framework, the feasibility set is written as
\[
\Theta=\bigcap_{(x,d)\in S}\{w\in\mathbb{C}^N:\ |d-x^T w|\le \alpha\},
\]
the instantaneous constraint set is
\[
H_n=\{w\in\mathbb{C}^N:\ |d(n)-x^T(n)w|\le \alpha\},
\]
and the membership set is
\[
\Psi_n=\bigcap_{i=1}^n H_i.
\]
The key operational consequence is that if the current error already satisfies the bound, there is no need to update. This gives set-membership algorithms their data-selective character [2605.18432].

Within that framework, affine projection introduces data reuse. Instead of relying only on the current regressor, the algorithm uses a block of recent data vectors. In the DS-CDMA treatment of set-membership adaptive algorithms, SM-AP is described as a generalized case of the SM-NLMS where \(P\) data vectors are used to increase the convergence speed. This places SM-AP between one-vector stochastic-gradient updates and least-squares or RLS-type recursions: it preserves the bounded-error philosophy of set-membership filtering while exploiting the fast convergence associated with projection over multiple recent regressors [1301.0097].

A central conceptual distinction follows from this formulation. Classical point-estimation adaptive filters seek a single point solution, whereas set-membership methods define a feasibility set of acceptable solutions and update only when the current estimate violates an error bound. For SM-AP, the affine projection machinery specifies how the new coefficient vector is selected inside, or on the boundary of, that feasibility region [2005.10124].

## 2. Canonical SM-AP recursion

In the linear real-valued formulation used for robustness analysis, the desired signal is modeled as
\[
d(k)=w_o^T x(k)+n(k),
\]
where \(w_o\) is the unknown optimal system, \(x(k)\) is the input vector, and \(n(k)\) is the noise or uncertainty term. The adaptive filter output is \(y(k)=x^T(k)w(k)\), the scalar estimation error is \(e(k)=d(k)-y(k)\), and the coefficient error vector is
\[
\widetilde{w}(k)=w_o-w(k).
\]
For projection order \(L\), the data matrix is
\[
X(k)=\big[x(k)\;x(k-1)\;\cdots\;x(k-L)\big],
\]
with associated error, desired-signal, noise, and constraint vectors collected over the same block [2005.10124].

The exact recursive SM-AP update is
\[
w(k+1)= \begin{cases} 
w(k)+X(k)A(k)\big(e(k)-\gamma(k)\big), & \text{if } |e(k)|>\gamma,\\[1mm]
w(k), & \text{otherwise},
\end{cases}
\]
where
\[
A(k)=\big(X^T(k)X(k)\big)^{-1},
\]
assuming \(X^T(k)X(k)\) is invertible. The same recursion can be written with the indicator function
\[
f(e(k),\gamma)= \begin{cases}
1, & \gamma < |e(k)|,\\
0, & \text{otherwise},
\end{cases}
\]
as
\[
w(k+1) = w(k)+X(k)A(k)\big(e(k)-\gamma(k)\big)\,f(e(k),\gamma).
\]
The update decision is therefore purely data selective: update if \(|e(k)|>\gamma\), and otherwise leave the coefficient vector unchanged [2005.10124].

In the complex DS-CDMA formulation, the same structure appears with a block observation matrix
\[
{\bf Y}[i] = [{\bf r}[i] ~\ldots~ {\bf r}[i-P+1]],
\]
and the practical SM-AP recursion becomes
\[
{\bf w}_{k}[i+1] = {\bf w}_{k}[i] + \mu_{w}[i] {\bf Y}[i] ({\bf Y}^{H}[i]{\bf Y}[i] + \delta {\bf I})^{-1} e_{k}[i]{\bf u},
\]
with
\[
\mu_{w}[i] = \left\{ \begin{array}{ll}
(1 - \gamma_k[i]/|e_{k}[i]|) & \textrm{if $|e_{k}[i]|>\gamma_k[i]$,}\\
0 & \textrm{otherwise.}
\end{array}\right.
\]
Here \({\bf u}=[1~0~\ldots~0]^T\), \(P\) is the projection order, and \(\delta\) is a small regularization constant [1301.0097].

## 3. Minimum-disturbance derivation and projection geometry

The projection interpretation of SM-AP is explicit. Whenever an update occurs, \(w(k+1)\) is the solution of the constrained optimization problem
\[
\min_{w(k+1)} \|w(k+1)-w(k)\|^2
\quad \text{subject to} \quad
d(k)-X^T(k)w(k+1)=\gamma(k).
\]
Thus, among all coefficient vectors satisfying the posterior error constraint, SM-AP picks the one with minimum Euclidean change from the current estimate. This minimum-disturbance principle is the defining geometric mechanism of the algorithm [2005.10124].

The DS-CDMA derivation makes the same structure explicit in block form:
\[
\begin{split}
{\textrm{minimize} & ~ || {\bf w}_k[i+1] - {\bf w}_k[i]||^2  \\
{\textrm {subject to} & ~ ({\bf b}_k[i] - {\bf Y}^H[i]{\bf w}_k[i+1]) = {\bf g}_k[i].
\end{split}
\]
Using Lagrange multipliers yields
\[
{\bf t}_{k}[i] = ({\bf Y}^{H}[i]{\bf Y}[i] + \delta {\bf I})^{-1} ({\bf e}_{k}[i]-{\bf g}[i]),
\qquad
{\bf w}_{k}[i+1] = {\bf w}_{k}[i] +  {\bf Y}[i] {\bf t}_{k}[i].
\]
To obtain the set-membership affine projection recursion, the boundary is chosen so that
\[
{\bf e}_{k}[i] - {\bf g}_{k}[i] = (e_{k}[i] - \gamma sgn(e_{k}[i])){\bf u}
= (1 - \gamma_k[i]/|e_{k}[i]|) e_{k}[i]{\bf u},
\]
while the \(P-1\) older a posteriori errors are kept constant. This is what turns the general constrained solution into the practical data-selective SM-AP update [1301.0097].

A common source of confusion is the word “projection.” The projection-operator literature distinguishes between projection of a point onto a set and modification of an update direction so that a parameter trajectory remains feasible. That distinction is directly relevant to adaptive filtering: the projection operator in adaptive systems is a direction-modification rule, not the classical Euclidean projector \(P_C(x)=\arg\min_{z\in C}\|x-z\|\). This suggests that SM-AP should be understood as a feasibility-preserving adaptive law built from projection geometry, rather than only as a matrix formula [1112.4232].

## 4. Stability, robustness, and non-divergence

The principal stability result currently available for the linear SM-AP algorithm is an \(\ell_2\)-stability analysis. Starting from the coefficient-error recursion, the paper derives the energy identity
\[
\|\widetilde{w}(k+1)\|^2 = \|\widetilde{w}\|^2 -\varepsilon^T A \varepsilon f +n^T A n f -2\gamma^T A n f +\gamma^T A\gamma f,
\]
and, at update times,
\[
\|\widetilde{w}(k+1)\|^2+\varepsilon^T A\varepsilon
=
\|\widetilde{w}(k)\|^2+n^T A n-2\gamma^T A n+\gamma^T A\gamma.
\]
This leads to the paper’s local robustness criterion: the one-step energy contracts if
\[
\gamma^T A\gamma < 2\gamma^T A n.
\]
If no update occurs, then
\[
\|\widetilde{w}(k+1)\|^2=\|\widetilde{w}(k)\|^2.
\]
These statements are the basis of the local robustness theorem [2005.10124].

Global robustness is obtained by summing the local inequalities over the update times. If the constraint vector is chosen so that
\[
\gamma^T(k)A(k)\gamma(k) < 2\gamma^T(k)A(k)n(k)
\]
for every updating iteration, then
\[
\frac{ \|\widetilde{w}(K)\|^2+\displaystyle\sum_{k\in K_{up}}\varepsilon^T(k)A(k)\varepsilon(k) }
{ \|\widetilde{w}(0)\|^2+\displaystyle\sum_{k\in K_{up}} n^T(k)A(k)n(k) }
\le 1.
\]
The paper also gives an existence result: if
\[
\gamma(k)=c\,n(k), \qquad 0<c<2,
\]
then the robustness condition always holds. The same paper immediately notes that this choice is not practical because \(n(k)\) is not observable [2005.10124].

The strongest claim of that analysis is distinct from global robustness. The paper argues that SM-AP never diverges, no matter how its parameters are chosen, because each active update solves a bounded posterior-error projection problem. In that reading, poor choices of the constraint vector may weaken robustness and worsen performance, but do not produce divergence. This distinction between conditional \(\ell_2\)-robustness and parameter-independent non-divergence is central to the theoretical interpretation of SM-AP [2005.10124].

## 5. Design parameters, adaptive bounds, and application-specific behavior

The main design parameters of SM-AP are the projection order, the error bound, and the regularization. The projection order \(P\) determines how many recent data vectors are reused; larger \(P\) increases convergence speed but also increases computational cost because the recursion contains a \(P\times P\) matrix inverse. The regularization \(\delta\) in
\[
({\bf Y}^{H}[i]{\bf Y}[i] + \delta {\bf I})^{-1}
\]
is included explicitly for improving robustness [1301.0097].

A major practical issue is specification of the error bound. In the DS-CDMA work, the fixed bound \(\gamma_k[i]\) is replaced by time-varying bounds. The Parameter Dependent Bound (PDB) is
\[
\gamma_k[i+1] = (1-\beta)\gamma_k[i] + \beta \sqrt{\alpha \|{\bf w}_k[i]\|^2 \hat{\sigma}_v^2[i]},
\]
and the Parameter and Interference Dependent Bound (PIDB) is
\[
\gamma_k[i+1] = (1-\beta) \gamma_k[i] + \beta \Big(\sqrt{ \tau ~{\hat v}^2[i]} + \sqrt{\alpha||{\bf w}_{k}||^{2} {\hat{\sigma}^{2}_{v}[i]}}\Big).
\]
Because the SM-AP step size is
\[
\mu_{w}[i] = 1-\frac{\gamma_k[i]}{|e_k[i]|}
\quad \text{when } |e_k[i]|>\gamma_k[i],
\]
a time-varying bound modifies both the update decision and the update magnitude. The same paper states that the SM-AP complexity is
\[
UR \times {\mathcal O}(PM + 2K_{inv}P^{2}),
\]
so the update rate \(UR\) directly scales the effective cost [1301.0097].

The reported DS-CDMA results attribute substantial gains to these adaptive bounds. For AP algorithms with \(P=3\), the PIDB mechanism gives the best BER performance across scenarios; the proposed PIDB mechanism with SM-AP and \(P=3\) can save up to \(3\ \mathrm{dB}\) compared to PDB and up to \(4\ \mathrm{dB}\) compared to SM-AP with fixed bounds for the same BER performance, and the PIDB approach can accommodate up to 4 more users than the PDB technique for the same BER. The interference-power estimator that supports PIDB tracks actual interference power very accurately, with average estimation error not exceeding about \(5\%\) of the estimated power level [1301.0097].

The 2020 stability paper complements those results with explicit update-rate measurements under different constraint-vector choices. In system-identification simulations, the mean update rates for SM-AP were \(32.7\%\) for the general constraint vector, \(9.8\%\) for SC-CV, and \(3.7\%\) for the noise CV. The same simulations reported that SM-AP with the practical SC-CV choice sharply reduced the number of iterations for which \(\|\widetilde{w}(k+1)\|^2 > \|\widetilde{w}(k)\|^2\), and that the idealized \(\gamma(k)=n(k)\) choice made \(\|\widetilde{w}(k)\|^2\) monotone decreasing [2005.10124].

## 6. Widely-linear and kernelized generalizations

A major recent generalization is the Augmented Set-Membership Affine Projection Algorithm (ASM-APA), developed for complex, second-order noncircular, and highly colored input signals. In that setting, the strictly linear model is insufficient, and the widely-linear filter output is
\[
y(n)=x^T(n)h(n)+x^H(n)g(n).
\]
The ASM-APA derivation combines the Augmented Affine Projection Algorithm (AAPA) with set-membership filtering, preserving the augmented affine projection geometry while updating only when the error exceeds a prescribed bound. The final recursion is
\[
h(n+1)=h(n)+\mu(n)e(n)X^*(n)\big(X^H(n)X(n)+X^T(n)X^*(n)+\delta I_P\big)^{-1}u_1,
\]
\[
g(n+1)=g(n)+\mu(n)e(n)X(n)\big(X^H(n)X(n)+X^T(n)X^*(n)+\delta I_P\big)^{-1}u_1,
\]
with
\[
\mu(n)= \begin{cases}
0, & |e(n)|\le \alpha,\\[1mm]
1-\dfrac{\alpha}{|e(n)|}, & |e(n)|>\alpha.
\end{cases}
\]
The paper emphasizes that ASM-APA reduces the average inversion burden by performing updates only on informative samples; it does not eliminate per-update inversion algebraically. In the reported experiments, ASM-APA consistently achieves smaller steady-state error than AAPA, its update rate remains around \(30\%\)–\(40\%\) in the highly correlated system-identification setting, and in stereophonic acoustic echo cancellation it attains faster convergence, lower steady-state inaccuracy, and an update rate of only \(0.52\%\) compared with AAPA [2605.18432].

A nonlinear generalization appears in the set-membership kernelized affine projection algorithm, denoted SM-KAP or SM-KAPA. In reproducing kernel Hilbert space form, the algorithm solves
\[
\min_{\boldsymbol{\omega}[i]} \left\| \boldsymbol{\omega}[i]-\boldsymbol{\omega}[i-1] \right\|^2
\quad \text{subject to} \quad
\boldsymbol{d}[i]-\boldsymbol{\Phi}^T[i]\boldsymbol{\omega}[i] = \bar{\boldsymbol{\gamma}}[i],
\]
which yields
\[
\boldsymbol{\omega}[i] = \boldsymbol{\omega}[i-1] + \boldsymbol{\Phi}[i] \left( \boldsymbol{\Phi}^T[i]\boldsymbol{\Phi}[i] \right)^{-1} \left( \eta[i]e[i] \right)\boldsymbol{u},
\]
with
\[
\eta[i] = \begin{cases}
1-\dfrac{\bar{\gamma}}{|e[i]|}, & |e[i]|>\bar{\gamma},\\[1ex]
0, & \text{Other Case}.
\end{cases}
\]
This is the RKHS counterpart of linear SM-AP: the linear data matrix is replaced by the feature-space matrix, and the correlation matrix is replaced by the kernel Gram matrix. In the reported time-series prediction experiments, SM-KAPA achieved test MSE \(0.0046603\) on Mackey-Glass and \(0.0029454\) on laser data, with \(K=7\), Gaussian kernel bandwidth \(1\), and error bound \(\sqrt{5}\sigma\) [1708.08142].

These variants preserve the same structural core: bounded-error feasibility, minimum-disturbance projection, and data-selective updating. What changes is the geometry of the regressor space—real linear, widely-linear augmented complex, or RKHS—together with the form of the projection matrix and the application domain [2605.18432][1708.08142].

Source: https://www.emergentmind.com/topics/set-membership-affine-projection-algorithm