Proportionate Recursive Maximum Correntropy Criterion Adaptive Filtering Algorithms and their Performance Analysis

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Proportionate Recursive Maximum Correntropy Criterion Adaptive Filtering

Algorithms and their Performance Analysis

Zhen Qina,b, Jun Taoc,∗, Le Yangd, Ming Jiange

aDepartment of Computer Science and Engineering, Ohio State University, Columbus, 43210, USA.

bSchool of Information Science and Engineering, Southeast University, Nanjing, 210096, China.

cKey Laboratory of Underwater Acoustic Signal Processing of the Ministry of Education,

School of Information Science and Engineering, Southeast University, Nanjing 210096, China.

dDepartment of ECE, University of Canterbury, Christchurch, 8020, New Zealand.

ePengcheng Lab, Shenzhen, 518066, China.

Abstract

The maximum correntropy criterion (MCC) has been employed to design outlier-robust adaptive ﬁltering algorithms,

among which the recursive MCC (RMCC) algorithm is a typical one. Motivated by the success of our recently proposed

proportionate recursive least squares (PRLS) algorithm for sparse system identiﬁcation, we propose to introduce the pro-

portionate updating (PU) mechanism into the RMCC, leading to two sparsity-aware RMCC algorithms: the proportionate

recursive MCC (PRMCC) algorithm and the combinational PRMCC (CPRMCC) algorithm. The CPRMCC is implemented

as an adaptive convex combination of two PRMCC ﬁlters. For PRMCC, its stability condition and mean-square perfor-

mance were analyzed. Based on the analysis, optimal parameter selection in nonstationary environments was obtained.

Performance study of CPRMCC was also provided and showed that the CPRMCC performs at least as well as the better

component PRMCC ﬁlter in steady state. Numerical simulations of sparse system identiﬁcation corroborate the advantage

of proposed algorithms as well as the validity of theoretical analysis.

Keywords: Adaptive ﬁlter, maximum correntropy criterion, proportionate updating, sparse system.

1. Introduction

The minimum mean square error (MMSE) and least squares (LS) criteria, have been widely adopted in the design of

adaptive ﬁltering algorithms, leading to e.g. the least mean squares (LMS) and recursive least squares (RLS) ﬁlters. These

classic algorithms have attained great success in many applications such as channel equalization [1], noise cancellation [2],

system modeling [3], just to name a few. However, the MMSE/LS criteria are known to be sensitive to outliers [4]. Outliers

are common in various contexts, such as sensor calibration [5] (due to sensor failure), face recognition [6] (owing to self-

shadowing, specularity, or brightness saturation), and others. In contrast, the emerging maximum correntropy criterion

(MCC) is more robust to outliers.

∗Corresponding authors.

Email addresses: qin.660@osu.edu (Zhen Qin), jtao@seu.edu.cn (Jun Tao), le.yang@cantebury.ac.nz (Le Yang),

jiangm@pcl.ac.cn (Ming Jiang)

Preprint submitted to Digital Signal Processing October 10, 2023

arXiv:2210.12311v2 [eess.SP] 8 Oct 2023

Many MCC adaptive ﬁltering algorithms have been developed [7, 8, 9, 10, 11, 12, 13]. They can be classiﬁed into two

categories: stochastic gradient type algorithms [7, 8, 9] and recursive MCC (RMCC) type algorithms [10, 11, 12, 13]. The

design of RMCC is somewhat similar to that of the RLS. In [7], an MCC algorithm was proposed for the environment with

impulsive noise and its theoretical excess mean square error (MSE) was derived using the energy conservation principle [8].

In [9], a generalized Gaussian density (GGD) cost function was employed to replace the Gaussian kernel used in original

MCC algorithm, leading to a generalized maximum correntropy criterion (GMCC) algorithm. Compared with stochastic

gradient type algorithms, the RMCC type algorithms can achieve superlinear convergence [10, 11, 12].

Motivated by the success of sparsity-aware MMSE/LS adaptive ﬁltering algorithms [14, 15, 16, 17, 18], the design of

sparse MCC adaptive ﬁltering algorithms have recently attracted great attentions [19, 20, 21, 22, 23]. In [19], a correntropy

induced metric (CIM) MCC (CIMMCC) algorithm has been designed by exploiting the CIM to approximate the l0norm

in the cost function. In [20, 21], the l1norm and a general regularization function have been respectively introduced to the

RMCC algorithm, leading to faster convergence under sparse systems. In [22, 23], the proportionate MCC (PMCC) and

hybrid-norm constrained PMCC algorithms were proposed for wide-sense sparse systems [24] and block sparse systems

[25], respectively.

In our recent work [26], the proportionate updating (PU) mechanism was introduced to the standard RLS, leading to the

proportionate recursive least squares (PRLS) algorithm with improved performance under sparse systems. This naturally

motivates us to incorporate a similar proportionate matrix to the existing RMCC to establish a proportionate recursive

maximum correntropy criterion (PRMCC) algorithm. According to the analysis in [27], the parameter θcontrolling the

trace of the proportionate matrix in the PRLS trades off between the initial convergence and steady-state performance.

Speciﬁcally, when θ≤N(Nis the ﬁlter length), the steady state of the PRLS is at least as good as the standard RLS.

However, its convergence rate may be slower. To achieve fast convergence and low steady-state error simultaneously, two

methods have been proposed for LMS-type algorithms: variable step approaches [28, 29, 30] and convex combination

[31, 32, 33, 34]. As θhas similar function as the step size in the LMS, we explore the feasibility of further improving the

PRMCC. In particular, we propose an adaptive convex combination of two PRMCC ﬁlters, leading to the combinational

PRMCC (CPRMCC) algorithm. Theoretical performance analysis of the PRMCC and CPRMCC are then provided. For

the PRMCC, its stability condition is ﬁrst derived based on the Taylor expansion approach. Then, we investigate its steady-

state performance and tracking ability, based on which theoretically optimal parameter values can be determined. For the

CPRMCC, we prove its performance is at least not worse than the better one of its two components. Simulation results on

sparse channel estimation are presented to demonstrate the effectiveness of proposed algorithms.

The rest of this paper is organized as follows. In Section 2, we develop the PRMCC and CPRMCC algorithms. Section 3

provides the stability condition and analyzes the mean-square performance for the PRMCC in stationary and nonstationary

environments. In addition, the superiority of CPRMCC is veriﬁed. In Section 4, simulation results are presented and

Section 5 concludes the paper.

Notation: We use bold capital letters, boldface lowercase letters, and italic letters, e.g. A,a, and ato, respectively,

represent matrix, vector, and scalar quantities. The superscript, (·)T, denotes the transpose, and E[·]represents the statistical

expectation. For a vector aof size N×1, its ln-norm is deﬁned as ||a||n= (PN

m=1 |am|n)1

n. The tr(A)returns the trace

of a matrix A.

2. Sparsity-aware RMCC Adaptive Filtering Algorithms

Consider a standard system identiﬁcation setting in the real domain with the following input-output relationship

d(n) = wT

NxN(n) + v(n),(1)

where wN= [w1, w2,··· , wN]Tis the N×1system impulse response vector to be estimated, and xN(n)=[x(n), x(n−

1),··· , x(n−N+ 1)]Tdenotes the input vector at time instant n. The d(n)and v(n)represent the system output and the

additive noise.

Given two random variables Xand Y, the correntropy is deﬁned as [35]

V(X, Y ) = E[κ(X, Y )] = Zκ(x, y)dFXY (x, y),(2)

where κ(·,·)is a shift-invariant Mercer kernel, and FXY (x, y)denotes the joint distribution function of (X, Y ). Unless

otherwise speciﬁed, the following Gaussian kernel is used in this work:

κ(x, y) = Gσ(e) = 1

√2πσ exp(−e2

2σ2),(3)

where e=x−yand σ > 0is the kernel bandwidth. The correntropy measures how similar two random variables are

within a neighborhood controlled by the kernel width σ. Speciﬁcally, the smaller the value of σ, the better it can eliminate

the impact caused by outliers. While σ→ ∞, such a metric will reduce to one global measure, i.e., mean square error

(MSE).

The RMCC employs the following cost function

JRMCC(n) =

i=1

λn−iexp(−e2(i|n)

2σ2),(4)

where 0<λ<1is a forgetting factor and

e(i|n) = d(i)−wT

N(n)xN(i).(5)

The weight vector wN(n)=[w1(n), w2(n),··· , wN(n)]Tdenotes an estimate of wNat time n. According to [21], the

weight update of RMCC is governed by

wN(n) = wN(n−1) + k(n)f(e(n|n−1)),(6)

where

f(e(n|n−1)) = exp(−e2(n|n−1)

2σ2)e(n|n−1),(7)

with the a priori error being e(n|n−1) = d(n)−wT

N(n−1)xN(n), and the (Kalman) gain vector k(n)is

k(n) = P(n−1)xN(n)

λ+ exp(−e2(n|n−1)

2σ2)xT

N(n)P(n−1)xN(n).(8)

The matrix P(n−1) can be iteratively computed using

P(n−1) = λ−1P(n−2) −exp(−e2(n−1|n−1)

2σ2)k(n−1)xT

N(n−1)P(n−2),(9)

so that (8) can be rewritten as

k(n) = λ−1P(n−1) −exp(−e2(n|n−1)

2σ2)k(n)xT

N(n)P(n−1)xN(n) = P(n)xN(n).(10)

2.1. The PRMCC Algorithm

Motivated by the PRLS from [26], an N×Nproportionate matrix G(n−1) is introduced into the update of (6), leading

to the PRMCC algorithm within the following update rule:

wN(n)=wN(n−1) + G(n−1)k(n)f(e(n|n−1)),(11)

where G(n−1) = diag{g1(n−1), g2(n−1), . . . , gN(n−1)}with the k-th diagonal element given by

gk(n−1) = θ(1 −α)

2N+ (1 + α)|wk(n−1)|

2||wN(n−1)||1+ϵ.(12)

Here, ϵis a small positive constant, α∈[−1,1), and θ > 0is the trace controller. The parameter θtrades off the

convergence and steady-state behaviors of the PRMCC. It is noted as σ→ ∞ in (7), the PRMCC algorithm reduces to the

PRLS algorithm.

The implementation of PRMCC algorithm is summarized in Algorithm 1.

2.2. The CPRMCC Algorithm

As mentioned before, the convergence and steady-state behaviors of the PRMCC is traded off by the parameter θin the

proportionate matrix. To simultaneously achieve fast convergence and good steady-state performance, we further propose

the CPRMCC algorithm, which is indeed an adaptive convex combination of two PRMCC ﬁlters with two different trace

Algorithm 1 The PRMCC Algorithm

Initialization: P(0) = δ−1IN,wN(0) = 0N×1,λ,δ,ϵ,θ,αand σare constants;

Iteration: For n= 1,2,···

e(n|n−1) = d(n)−wT

N(n−1)xN(n)

k(n) = P(n−1)xN(n)

λ+exp(−e2(n|n−1)

2σ2)xT

N(n)P(n−1)xN(n)

gk(n−1) = θ(1−α)

2N+θ(1 + α)|wk(n−1)|

2||wN(n−1)||1+ϵ, k = 1,2,...N

G(n−1) = diag{g1(n−1), g2(n−1), ..., gN(n−1)}

f(e(n|n−1)) = exp(−e2(n|n−1)

2σ2)e(n|n−1)

wN(n) = wN(n−1) + G(n−1)k(n)f(e(n|n−1))

P(n) = λ−1[P(n−1) −exp(−e2(n|n−1)

2σ2)k(n)xT

N(n)P(n−1)]

controllers. The output of the CPRMCC is given as

y(n) = ρ(n)wT

1(n)xN(n) + (1 −ρ(n))wT

2(n)xN(n)

=wT

N(n)xN(n),

(13)

where w1(n)and w2(n)are updated as (11) with trace controllers being set to θ1and θ2. The overall weight estimate is

then

wN(n) = ρ(n)w1(n) + (1−ρ(n))w2(n).(14)

The mixing parameter ρ(n)is deﬁned as a sigmoidal function

ρ(n) = sgm[b(n−1)] = 1

1 + exp(−b(n−1)),(15)

where b(n−1) is iteratively updated via [34]

b(n−1) = b(n−2) + µbexp(−e2(n−1|n−1)

2σ2

)e(n−1|n−1)(y1(n−1)−y2(n−1))ρ(n−1)(1 −ρ(n−1))b+

−b+

,(16)

in which

y1(n) = wT

1(n)xN(n)and y2(n) = wT

2(n)xN(n),(17)

and µband σbare, respectively, the step size and kernel bandwidth. The values of b(n−1) has been limited to the interval

[−b+, b+][31, 32, 33] to avoid possible pre-mature termination of the update when ρ(n−1) is close to zero or one.

The detailed implementation of CPRMCC algorithm is summarized in Algorithm 2.

The CPRMCC can be further improved by speeding up the convergence of the slower component PRMCC ﬁlter via the

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ProportionateRecursiveMaximumCorrentropyCriterionAdaptiveFilteringAlgorithmsandtheirPerformanceAnalysisZhenQina,b,JunTaoc,∗,LeYangd,MingJiangeaDepartmentofComputerScienceandEngineering,OhioStateUniversity,Columbus,43210,USA.bSchoolofInformationScienceandEngineering,SoutheastUniversity,Nanjing,210096...

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