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Adaptive Projected Subgradient Method and

its Applications to Robust Signal Processing

(Invited Paper)

Isao Yamada, Konstantinos Slavakis, Masahiro Yukawa and Renato L. G. Cavalcante

Dept. of Communications and Integrated Systems, Tokyo Institute of Technology, Tokyo 152-8552 JAPAN

Emails: {isao,slavakis,masahiro,renato}@comm.ss.titech.ac.jp

Abstract—The adaptive projected subgradient method of-

fers a unified mathematical perspective for the adaptive (set-

membership / set-theoretic) filtering schemes. In this paper,

we introduce an overview of its recent theoretical advances

and successful applications to robust signal processing problems

including the stereo acoustic echo canceling, the MAI suppression

in DS/CDMA receivers, and the robust adaptive beamforming

with array antenna systems.

I. INTRODUCTION

A common strategy among many schemes ([1–7] and refer-

ences therein) in adaptive set-membership filtering / adaptive

set-theoretic filtering is the iterative approximation of an esti-

mandum as a point in the intersection of possibly time-varying

family of closed convex sets. Such closed convex sets are

defined, to restrict candidates of the estimandum, with mea-

surements of signals or with a priori knowledge1. The great

flexibility of closed convex sets brings significant benefits to

us in realizing advanced adaptive filtering in accordance with

the intended applications (see the Set-Membership Normalized

LMS (SM-NLMS) [4], the frequency-domain Set-Membership

Normalized LMS (F-SM-NLMS) [7] and the Adaptive Parallel

Subgradient Projection (Adaptive PSP)[5]).

The above mentioned strategy has obvious commonality

with algorithmic solutions to the convex feasibility problems

(see for example [8–12] and references therein), hence the

goal is achieved mathematically by minimizing a family of

nonnegativeconvexobjectives; e.g., distances to closed convex

sets, in a real Hilbert space. (Note: the complex adaptive

filtering problems can be formulated naturally in a real space).

On the other hand, an inherent difference between the

scenarios for the convex feasibility problems and those for

the adaptive filtering problems must be that any algorithm

for the convex feasibility problem is based on infinitely many

uses of information on a specific nonnegativeconvexobjective

but any algorithm for the adaptive filtering problems can not.

This is because the convex objective for the adaptive filtering

problems keeps changing due to the (possibly nonstationary)

input and noise processes.

The Adaptive Projected Subgradient Method (APSM) [13–

15] was developed as an efficient algorithm for asymptotic

minimization of a certain sequence of nonnegative convex

functions. The APSM is a natural extension of the Polyak’s

subgradient algorithm [16], for nonsmooth convex optimiza-

tion problem with a fixed target value, to the case where the

convex objective itself keeps changing in the whole process.

It is revealed that the APSM offers a useful mathematical

foundation of a wide range of the projection based adaptive

1Similar idea is found in a fundamental principle in Husserl’s phenomenol-

ogy [Edmund Husserl (1859-1938)].

filtering algorithms. Indeed, by designing a certain sequence

of convex objectives, a variety of adaptive filtering algorithms

are derived in a unified manner as simple examples of the

adaptive projected subgradient method.

In this paper, we introduce basic idea2of the APSM and

its successful applications to robust signal processing problems

including the stereo acoustic echo canceling [17,18], the MAI

suppression in DS/CDMA receivers [19–21], and the robust

Capon beamforming with array antenna systems [22].

II. ADAPTIVE PROJECTED SUBGRADIENT METHOD

A. Preliminaries

Let H be a (possibly infinite dimensional) real Hilbert space

equipped with an inner product ?x,y?, ∀x,y ∈ H, and its in-

duced norm ?x? := ?x,x?1/2, ∀x ∈ H. A set C ⊂ H is called

convex if ∀x,y ∈ C, ∀ν ∈ (0,1), νx + (1 − ν)y ∈ C. For

any nonempty closed convex set C ⊂ H, the metric projection

PC: H → C maps x ∈ H to the unique vector PC(x) ∈ C

such that d(x,C) := miny∈C?x − y? = ?x − PC(x)?. A

function Θ : H → R is said to be convex if ∀x,y ∈ H and

∀ν ∈ (0,1), Θ(νx + (1 − ν)y) ≤ νΘ(x) + (1 − ν)Θ(y). If

Θ is continuous and convex, lev≤0Θ := {x ∈ H | Θ(x) ≤

0} is closed convex and the subdifferential of Θ at y (the

set of all the subgradients of Θ at y) satisfies ∂Θ(y) :=

{s ∈ H | ?x − y,s? + Θ(y) ≤ Θ(x),∀x ∈ H} ?= ∅. By this

definition, we have 0 ∈ ∂Θ(y) ⇔ Θ(y) = minx∈HΘ(x).

The convex function Θ : H → R has a unique subgradient at

y ∈ H if Θ is differentiable at y. This unique subgradient is

nothing but the gradient ∇Θ(y), i.e., ∂Θ(y) = {∇Θ(y)}.

B. Adaptive Projected Subgradient Method

A primitive idea of the set-membership adaptive filtering /

the set-theoretic adaptive filtering is found in the Normalized

Least Mean Squares (NLMS)[1]. The NLMS approximates it-

eratively a possibly time-varying estimandum, which is highly

expected to belong to (or expected to exist closely to) the

intersection of time-varying sequence of hyperplanes in a

vector space, by computing the metric projection onto each

hyperplane in sequential order3. Obviously, at each time, the

task of the NLMS is to minimize a nonnegative convex

function defined as the distance to the current hyperplane.

This simple strategy of the NLMS has been proven very

successful through extensive real-world applications dealing

with stationary / nonstationary random signals. The excellent

2For complete discussion and many examples of the APSM, see [15]

appeared in a more mathematical publication.

3To accelerate the speed of convergence of the NLMS, the APA [2,3]

introduces the metric projection onto a linear variety defined as the intersection

of certain number of hyperplanes.

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robust performance of the NLMS was reaffirmed through H∞-

theory [23]. The APSM is a natural generalization of the

NLMS and designed to minimize asymptotically a sequence

of more general nonnegative convex functions than the simple

distance to hyperplane. The next theorem presents the APSM

and its elementary but useful properties.

Theorem 1: (APSM [13–15]) Let Θk: H → [0,∞) (∀k ∈

N) be a sequence of continuous convexfunctions and K ⊂ H a

nonempty closed convex set. For an arbitrarily given h0∈ K,

the APSM produces a sequence (hk)k∈N⊂ K by

where Θ?

Then the sequence (hk)k∈Nsatisfies the followings4.

(a) (Monotone approximation)Suppose that

hk+1:=

PK

hkotherwise,

?

hk− λk

Θk(hk)Θ?

?Θ?

k(hk)

k(hk)?

2

?

if Θ?

k(hk) ?= 0,

(1)

k(hk) ∈ ∂Θk(hk) and 0 ≤ λk≤ 2.

hk?∈ Ωk:= {h ∈ K | Θk(h) = Θ∗

where Θ∗

?

(∀h∗(k)∈ Ωk)

(b) (Boundedness, Asymptotic optimality) Suppose ∃N0 ∈

N s.t. Θ∗

∅. Then (hk)k∈N is bounded. If we use ∀λk

[ε1,2 − ε2] ⊂ (0,2), we have lim

vided that (Θ?

Example 1: Let S(k)

ι

⊂ H (ι ∈ Ik ⊂ Z) and K ⊂ H, be

nonempty closed convex sets. Define a sequence of convex

functions by Θk(h) :=

Lk

ι∈Ik

∀k ∈ N, where?

Then we have Θ?

Lk

∈ ∂Θk(hk) if Lk ?= 0, and Θ?

otherwise. By applying (1) to Θkand K ⊂ RN, we deduce a

scheme:

?

ι∈Ik

where h0∈ K, µk∈ [0,2M(1)

Obviously this is a generalization of Algorithm 1 in [5], hence

it includes NLMS [1], APA [2,3] and the adaptive parallel

k} ?= ∅,

k:= infh∈KΘk(h). Then, by using ∀λk ∈

1 −

?hk+1− h∗(k)? < ?hk− h∗(k)?.

0,2

?

Θ∗

k

Θk(hk)

??

, we have

k= 0, ∀k ≥ N0 and Ω :=

?

k≥N0Ωk ?=

∈

k→∞Θk(hk) = 0 pro-

k(hk))k∈Nis bounded.

2

1

?

ω(k)

ι d(hk,S(k)

ι

)d(h,S(k)

ι

),

ι∈Ikω(k)

ι ) ?= 0, and Θk(h) := 0 otherwise.

k(hk) =

ι

k(hk) = 0 ∈ ∂Θk(hk)

ι

= 1, {ω(k)

ι }ι∈Ik⊂ (0,1], if Lk:=

?

?

ι∈Ikω(k)

ι d(hk,S(k)

1

?

ι∈Ikω(k)

hk− PS(k)

ι

(hk)

?

hk+1:= PK

hk+ µk

??

k] and

????PS(k)

PS(k)

ι

otherwise.

ω(k)

ι PS(k)

ι

(hk) − hk

??

,

M(1)

k

:=

?

ι∈Ikω(k)

?

1,

ι

ι

(hk)−hk

(hk)−hk

????

2

2, if hk / ∈?

????

ι∈Ikω(k)

ι

????

ι∈IkS(k)

ι ,

4Recently the metric projection PKin (1) was extended to arbitrary strongly

attracting nonexpansive mapping T [24]. By this generalization, we can

minimize asymptotically the sequence of nonnegative convex functions Θk

(∀k ∈ N) over the fixed point set Fix(T) := {x ∈ H | T(x) = x}.

For example, if we define T :=?m

i=1wiPKi(wi> 0, i = 1,2,...,m,

i=1PKifor closed convex sets Ki(i = 1,2,...,m)

i=1Ki?= ∅, we have Fix(T) =?m

?m

i=1wi= 1) or?m

s.t.?m

i=1Ki.

subgradient projection (Adaptive PSP) [5] as its special ex-

amples. The Adaptive PSP uses weighted average of multiple

subgradient projections to keep low computational cost of the

NLMS as well as to achieve fast and stable convergence even

in severely noisy environment.

Example 2: (Embedded constrained version) Let V := v+

M be a nonempty linear variety, where v ∈ H, M ⊂ H is

a closed subspace of H, and Θk: V → [0,∞) a continuous

convex function. Then Ψk: H ? h ?→ Θk(PV(h)) ∈ [0,∞)

satisfies Ψk(h) = Θk(h), ∀h ∈ V and PM(Θ?

∂Ψk(h), ∀h ∈ H, ∀Θ?

Ψk and K := V yields a new algorithm for asymptotic

minimization of (Θk)k∈Nover V :

?

hkotherwise,

k(PV(h))) ∈

k∈ ∂Θk(h). Application of (1) to

hk+1:=

hk− λk

Θk(hk)PM(Θ?

?PM(Θ?

k(hk))

k(hk))?

2

if Θ?

k(hk) ?∈ M⊥,

where h0∈ V , Θ?

algorithm ensures (hk)k∈N⊂ V . The constrained NLMS [6] is

an example of the embedded constrained version for Θk(h) :=

d(h,Vk) where Vkis a hyperplane.

k(hk) ∈ ∂Θk(hk) and 0 ≤ λk ≤ 2. This

III. APPLICATIONS

A. Stereophonic Acoustic Echo Canceling

Multi-microphone systems have been in the spotlight in

place of single-microphone systems in many applications such

as hands-free mobile telephony and teleconferencing, since

the performance of single-microphone systems is generally

limited. To realize high quality in the communication, it is

hence a central burden to develop an efficient and stable echo

canceling scheme. Main difficulties of multi-channel acoustic

echo cancellation (MAEC) are as follows: (i) strong cross-

correlation among signals observed at the microphones in

the transmission room causes slow convergence [25], (ii) a

large number of filter taps per each microphone [(# of loud-

speakers) × (a few thousand)] are required for sufficient echo

cancellation due to long acoustic impulse response stemming

from slow propagation of sounds, (iii) there may exist high

background noise and (iv) speech signals have generally high

auto-correlation and nonstationarity. Therefore, the classical

algorithms, NLMS, APA and RLS, are not suitable for MAEC

due to (i)·(iv) for NLMS, (ii)·(iii) for APA, and (ii)·(iv) for

RLS.

Recently, an efficient technique named pairwise optimal

weight realization (POWER) has been proposed [26], which

realizes reasonable weights in the adaptive PSP (see Example

1). Based on this weighting technique, an efficient stereo-

phonic echo canceling scheme, which can be readily extended

to multi-microphone systems, has also been proposed [18].

The scheme in [18] is compared to NLMS and APA in Fig. 1

under SNR = 25 dB to find h∗∈ R2000with speech input

signals. For all methods, we employ a nonlinear preprocessing

technique to decorrelate the input signals. We observe that the

proposed scheme is approximately twice faster than APA.

B. MAI Suppression for DS/CDMA systems

In DS/CDMA systems, as users share the same frequency

band at the same time, the signal originated from a single user

is affected by the interference originated from other users at

the receiver side [27,28]. Conventional receivers, which are

essentially matched filters, completely ignores the structure of

270

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0 20 40 60

-30

-20

-10

0

System Mismatch (dB)

Time (sec.)

NLMS

APA

Proposed

Fig. 1.System mismatch curve: Proposed versus NLMS and APA.

this multiple-access interference (MAI) and thus yields a bit-

error-rate (BER) that is usually much worse than the minimum

achievable probability of error [28]. Therefore, a great deal

of effort has been devoted to developing linear detectors

that present good performance and manageable complexity

[29,30]. Basically, these detectors are adaptive linear filters

that minimize either the mean-square-error (MSE) between

the desired output and the filter output, or the filter output

energy subject to a constraint that is determined by the desired

user’s signature. In order to track the optimal solution of

these two minimization problems, set-theoretic receivers can

be applied [19–21,31,32] (Excellent performance realized by

the set-membership filtering approach was also confirmed in

its applications [33,34] to the multiuser detection and the

adaptive beamforming).

The APSM is a set-theoretic approach that gives the de-

signer the option to use as much information as available

at each iteration. Therefore, fast convergence rate and good

steady-state performance can be achieved at the same time

[19–21] (see also Fig. 2). The main idea of the receivers based

on this method is as follows: for each received symbol, sets

that describe desired characteristics of an optimal filter are

built (NOTE: explicit knowledge of the transmitted symbol

may not be necessary), and then the current filter is projected

onto the available sets, which may include sets based on

past transmitted symbols. The additional complexity gained by

dealing with more data at each iteration can be alleviated by

using parallel computing and/or by calculating low-complexity

approximations of the true projections. By exploiting geomet-

rical properties of the projections onto closed convex sets, we

have shown [20] that it is possible to achieve remarkably good

convergence speed even when the receiver uses a moderate

number of symbols at the same time.

More recently, the results in [19,20] have been extended to

cope with a variety of assumptions about the channel model

and the available information at the receiver side [21]. The

blind receivers described in this work have lower complexity

and better BER for fast time-varying flat fading channels as

compared to some algorithms that require a matrix inversion.

It is also shown that many receivers based on the MSE

cost function or the output energy cost function are simple

examples of the adaptive projected subgradient method.

C. Robust Capon Beamforming

The Capon Beamformer (CB) is a celebrated solution to

the problem of finding w ∈ argminz∈CN, s∗

∀k ∈ N, where Ry(k) := E{y(k)y∗(k)}, k ∈ N, (E{·}

0z=1z∗Ry(k)z,

05001000 15002000

−15

−10

−5

0

5

10

15

Iteration number

Output SINR (dB)

NLMS (step size=0.6)

RLS

Proposed

(number of symbols used at each iteration=128,

step size=0.2Mn,

forgetting factor for amplitude estimation=0.01)

GP (step size=0.6)

Gold sequences of length 31, SNR=15 dB, 6 users

Interfering users have 20 dB power advantage

Fig. 2.

linear filters. The proposed (blind) method based on the APSM [19] has only

knowledge about the desired user’s signature. The Generalized Projection (GP)

algorithm [31] knows the desired user’s amplitude and signature. The RLS

and NLMS algorithms use training sequences.

Output signal to interference-noise ratio (SINR) curves for different

denotes expectation, ∗ stands for complex conjugate trans-

position) is the correlation matrix

(y(k))k∈N⊂ CNreceived by an array of N elements. Given

k ∈ N, the minimizer that ’rejects’ interference and noise

by the mapping y(k) ?→ w∗y(k) under the assumption that

the Signal Of Interest (SOI) suffers no distortion is the array

weighting vector wCB(k) =

s∗

vector s0gathers all the spatial characteristics of the array as

well as the SOI Direction Of Arrival (DOA).

Unfortunately, CB is sensitive to errors like DOA mis-

matches, poor array calibration, unknown sensor mutual cou-

pling [35], which cause erroneous estimates of s0. Unlike the

classical Diagonal Loading (DL) method where for a fixed

empirical value of ?DL > 0 the matrix ?DLIN (IN is the

identity matrix in CN×N) is added to Ry(k) prior its inversion

∀k ∈ N, very recent research [35] devises iterative methods

to calculate the optimal (in some sense) DL parameter by

explicitly using the uncertainty about the SOI steering vector.

In [22], we follow a different path from DL techniques and

we design a robust CB by the APSM. First, a characterization

of a data-independent (independent from (y(k))k∈N) robust

beamformer w ∈ CNis given as a time-varying convex

feasibility problem in the R2N+3space: the set of all robust

beamformers against SOI steering vector errors is the intersec-

tion of a finite number of time-independenticecream cones and

a finite number of time-dependent hyperplanes. Furthermore,

in order to employ the received data in the design, we generate

a sequence of halfspaces that most likely contain the desired

beamformer. Finally, we use Example 1 where instead of the

mapping PK we have used the composition of the metric

projection mappings onto the icecream cones; a strongly

attracting nonexpansive mapping. This algorithmic solution

becomes a special case of the extension [24] (See the footnote

given with Theorem 1).

Five interferences arrive at a Uniform Linear Array (ULA),

N = 20, each one of power σ2

3, ’Ideal’ stands for CB with no uncertainty, ’CB-DL’ for

its DL version, and ’SOCP’ for the approach in [35]. The

of the random process

Ry(k)−1s0

0Ry(k)−1s0. The SOI steering

j, j = 1,...,5. In Fig.

271

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−5 051015 2025 303540455055

10

12

14

16

18

20

22

SINR (dB)

INR (dB)

CB−DL

SOCP

APSM

Ideal

Fig. 3.

where sj is the steering vector of the j-th jammer, j = 1,...,5, σ2

the SOI power, and σ2

INR :=?5

Define SINR = (σ2

0|s∗

0w|2)/(?5

j=1σ2

j|s∗

jw|2+ σ2

n?w?2),

0is

nis the noise power. We set SNR = 10dB. Let

n.

j=1σ2

j/σ2

results exhibit the excellent performance of the APSM in

cases where the DL techniques face difficulties, i.e. where

the INR is moderately larger than SNR or in other words in

SOI-contaminated situations.

Acknowledgement: The authors would like to thank Prof. Ko-

hichi Sakaniwa of Tokyo Institute of Technology for helpful

discussions and comments. The 1st author also would like to

thank Prof. Yih-Fang Huang of University of Notre Dame for

his kind invitation to ISCAS2006. This work was supported

in part by JSPS Grants-in-Aid (C-17500139).

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IEICE

“Efficient fast stereo

IEICE

“Efficient blind

“Set-theoretic

“Efficient robust adaptive

“Blind adaptive multiuser

“Adaptive

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