COMBINING BLIND SOURCE EXTRACTION WITH JOINT APPROXIMATE
DIAGONALIZATION: THIN ALGORITHMS FOR ICA
Se gio C uces
Signal P ocessing G oup
Ingenie os, Camino Descub imien os,
41092-Se ille, Spain.
h p:// ien o.us.es/˜se gio
se [email p o ec ed]
And zej Cichocki
B ain Science Ins i u e
RIKEN, 2-1 Hi osawa, Wako-shi,
Sai ama, 351-0198 Japan.
h p://www.bsp.b ain. iken.go.jp/
cia@b ain. iken.go.jp
ABSTRACT
In his pape a mul i a ia e con as unc ion is p oposed
o he blind signal ex ac ion o a subse o he indepen-
den componen s om a linea mix u e. This con as com-
bines he obus ness o he join app oxima e diagonaliza-
ion echniques wi h he lexibili y o he me hods o blind
signal ex ac ion. I s maximiza ion leads o hie a chical and
simul aneous ICA ex ac ion algo i hms which a e espec-
i ely based on he hin QR and hin SVD ac o iza ions.
The in e es ing simila i ies and di e ences wi h o he exis -
ing con as s and algo i hms a e commen ed.
1. INTRODUCTION
In he las decades powe ul c i e ia and algo i hms has been
de eloped o sol e he p oblem o he analysis o he inde-
penden componen s in a linea mix u e [1, 2, 3]. In he
his o y o his p oblem one may dis inguish o di e en ap-
p oaches: he i s one is usually named blind signal sepa-
a ion (BSS) and consis s in he simul aneous es ima ion o
all he la en independen componen s om he mix u e; he
second one is known as blind signal ex ac ion (BSE) and
consis s in he es ima ion o only a subse o he indepen-
den componen s.
The BSE p oblem can be ega ded as mo e gene al and
lexible han BSS because o he ollowing wo easons: 1)
BSE includes BSS as he pa icula he case whe e one is
in e es ed in all he independen componen s, 2) BSE has
compu a ional ad an ages o e BSS i only a small subse
o he independen componen s is o in e es . These compu-
a ional sa ings can be impo an in applica ions like MEG
and EEG, whe e he decomposi ion o he obse a ions con-
side s a la ge numbe o possible independen componen s
bu only a ew o hem a e o in e es .
This wo k has been suppo ed by he CICYT p ojec o he Go e nmen
o Spain (G an TIC2001-0751-C04-04).
The o iginal c i e ia o blind signal ex ac ion we e ba-
sed in he hie a chical he eco e y o he independen com-
ponen s [4, 5] one by one al e na ing ex ac ion wi h de-
la ion. These c i e ia ha e been ex ended o allow he si-
mul aneous ex ac ion o an a bi a y numbe o indepen-
den componen s [6, 7]. Ano he desi ed p ope y o any
BSS/BSE c i e ion is i s obus ness, unde s ood in he sense
o i s abili y o gi e accu a e es ima es om he a ailable
da a. Popula echniques o blind signal sepa a ion [8, 9,
10] a e obus in he p e ious sense, ha ing c i e ia based
on he join app oxima e diagonaliza ion o se e al cumu-
lan slices. Howe e , up o ou knowledge, no equi alen
app oach o blind signal ex ac ion has been ob ained ye .
Thus, he pu pose o his con ibu ion is o p esen algo-
i hms ha pe o m he simul aneously ex ac ion a subse
o independen componen s om he mix u e by combining
he in o ma ion o se e al cumulan slices o he obse a-
ions p ocess.
2. SIGNAL MODEL
As shown in igu e 1, he chosen signal model o he ob-
se a ions x( ) = [x1( ),··· , xM( )]Tobey he ollowing
equa ion
x( ) = As( ) + n( )(1)
whe e s( ) = [s1( ),··· , sN( )]Tis he signal ec o p o-
cess o Nindependen componen s, n( ) he noise ec o
p ocess, and Ais a M×Nmixing ma ix. We conside he
ollowing assump ions:
A1 The componen s o s( )a e mu ually independen , lo-
cally s a iona y and no malized o ze o mean and uni
a iance.
A2 The noise ec o p ocess n( )is locally s a iona y, Gaus-
sian, whi e (Rn( 2, 1) = δ( 2− 1)E[n( )(n( ))H])
and wi h a known co ela ion ma ix Rn( , ) =
463
4 h In e na ional Symposium on Independen Componen Analysis and Blind Signal Sepa a ion (ICA2003), Ap il 2003, Na a, Japan
Axy
U
N
+
n
MNP
W
sz
G
Fig. 1. Signal model o he blind ex ac ion o Psou ces.
E[n( )(n( ))H]o one ha can be accu a ely es i-
ma ed om he obse a ions, pe haps using ac o ana-
lysis o any obus p ewhi ening echnique.
A3 The mixing ma ix Ais ull-column ank.
A4 Fo a gi en subse o Pindependen componen s
{s1( ),...,sP( )}, ha one wishes o ex ac , he e
exis a se Ω = {τk= ( (k)
1,..., (k)
q), k = 1,..., :
τk∈Rqi q > 2, τk∈R2 {( , ),∀ ∈R}i
q= 2}and a pe mu a ion σo he indices 1,...,P
ha so s he ollowing s a is ic o he componen s
ψΩ(sj) = X
τ∈Ω
wτ|Cum(sj( 1),··· , sj( q))|2
in such a way ha hese inequali ies hold ue
ψΩ(sσ(1))>···> ψΩ(sσ(P))>
ψΩ(sP+1)≥ · · · ≥ ψΩ(sN)(2)
F om A1 one ob ains AAH=Rx( , )−Rn( , ). F om
A2 and A3, using p incipal componen analysis one can
p ojec he obse a ions on o he signal subspace o educe
hei dimensionali y om M o Nand also sphe e he e-
sul ing signals. The N×Mp ewhi enning sys em W=
(AAH)−1/2gi es he ec o o p ep ocessed obse a ions
z( ) = Wx( )(3)
In o de o ex ac Pindependen componen s (1≤P≤
N) om he mix u e we use a P×Nma ix Uwhich
is semi-uni a y (UUH=IP). This ma ix mul iplies he
p ep ocessed obse a ions o gi e he ou pu s o es ima ed
sou ces y( ) = [y1( ),··· , yP( )]Tas
y( ) = Uz( ) = Gs( ) + UWn( )(4)
whe e G=UWA deno es he global ans e unc ion
om he independen componen s o he ou pu s.
Assump ion A4 gua an ees ha he e exis and o de e-
la ion among he sou ces ha is maximized by hose we
wan o ex ac .
3. EXTRACTION OF A SINGLE SOURCE
We will i s analyze he case o P= 1, i.e., he ex ac ion
o a single sou ce. Then Uand Ga e ow ec o s bo h o
uni 2-no m and he e is a single ou pu y( ) = Uz( ). We
p opose o es ima e he desi ed independen componen by
join ly maximizing a weigh ed squa e sum o cumulan s o
ixed1o de q≥2, de e mined by he uples τ= ( 1,··· ,
q)con ained he se Ω. A con as unc ion ha achie es
his objec i e is gi en by
ψΩ(y) = X
τ∈Ω
wτ|Cum (y( 1),··· , y( q))|2
subjec o kUk2= 1.(5)
we e wτa e posi i e weigh ing e ms. The chosen no a ion
1,..., qin (5) is due o he ac ha om A1 and A2 he
obse a ion p ocess may be la ge e m non-s a iona y, case
o which he con as can exploi he same cumulan slices
a di e en imes (using segmen s o quasi-s a iona y da a).
The p oblem wi h his app oach is in he di icul y o he
op imiza ion o (5), which is highly non-linea wi h espec
o U. The ollowing heo em, whose p oo is ske ched in
he Appendix, shows how o ci cum en his di icul y by
p oposing a simila con as unc ion o (5) bu whose de-
pendence wi h espec o each o he ex ac ing sys em can-
dida es is quad a ic, and hus, much mo e easy o op imize
using algeb aic me hods.
Theo em 1 Unde he assump ions A1-A4, pa icula ized
o he ex ac ion o only one independen componen s1,
he e exis a se Ω o which
ψΩ(s1)> ψΩ(sj)∀j= 2,...,N. (6)
Conside ing a se o qcandida es o he ex ac ing sys em
{U[1],...,U[q]}and he se o hei espec i e ou pu s ¯y=
{y[1],...,y[q]}, he ollowing mul i a ia e unc ion
ψΩ(¯y) = X
τ∈Ω
wτ

Cum(y[1]( 1),··· , y[q]( q))


2
subjec o kU[m]k2= 1, m = 1,...,q.(7)
whe e wτ>0, is a con as unc ion whose global maxi-
mum leads o he ex ac ion o he desi ed sou ce, i.e., a
his ex eme poin y[1]( ) = ···=y[q]( ) = s1( ).
We should no e ha his con as admi s a leas squa es ma ch-
ing in e p e a ion associa ed wi h he ank one app oxima-
ion o cumulan enso s [11]. A good me hod o maxi-
mize he p oposed con as ψΩ(¯y)is o op imize i cycli-
cally wi h espec o each one o he elemen s U[m], m =
1,...,q, while keeping ixed he o he s. In he ollowing,
he supe index ()[k]will con inue deno ing he k- h a i-
able (cyclic no a ion) while he supe index ()(k)will indi-
ca e he a iable aken alue a he k- h i e a ion (sequen-
ial no a ion). Then, no e ha a he (k) h i e a ion he [(k
1The esul s o he pape also apply o an a bi a y combina ion o cu-
mulan s wi h di e en o de s bu , due o he somewha mo e cumbe some
no a ion i needs, his ex ension will be p o ided elsewhe e.
464
mod q)+1] a iable will be op imized. Due o he in a ian
p ope y o ψΩ(¯y)wi h espec o pe mu a ions in i s a gu-
men s, he cyclic maximiza ion o he con as is equi alen
o he sequen ial maximiza ion o he unc ion
φΩ(U(k)) = X
τ∈Ω
wτ|Cum(y(k)( 1), y(k−1)( 2),···
...,y(k−q+1)( q))|2
=U(k)M(k−1)U(k)H(8)
wi h espec o he ex ac ion sys em U(k) h ough i e a-
ions. No e ha M(k−1) is a cons an ma ix (as long as
U(k−1),··· ,U(k−q+1) a e kep ixed) gi en by
M(k−1) =X
τ∈Ω
wτc(k−1)
zy(τ)c(k−1)
zy(τ)H
(9)
c(k−1)
zy(τ) = Cum(z( 1), y(k−1)( 2),··· , y(k−q+1)( q))
Since he ec o U(k)Hwhich maximizes φ(U(k))is he
eigen ec o associa ed o he dominan eigen alue o M(k−1),
we can mo e he ex ac ing ec o U(k−1) owa ds he solu-
ion wi h one o mo e i e a ions o any o he s anda d eigen-
pai inding me hods. S a ing om he p e ious solu ion, i
one conside s o use Li e a ions o he powe me hod o
app oxima e he dominan eigen ec o (in p ac ice L= 1
wo ks well), he ollowing ex ac ion algo i hm is ob ained
U(0) =U(k−1)
FOR l= 1 : L
U(l)=Pτ∈Ωwτd(l−1)
y(τ)c(k−1)
zy(τ)H



Pτ∈Ωwτd(l−1)
y(τ)c(k−1)
zy(τ)H


2
(10)
END
U(k)=U(L)
whe e d(l−1)
y(τ) = U(l−1)c(k−1)
zy (τ).
3.1. Con e gence analysis
The i e a i e op imiza ion o he unc ion φΩ(·)wi h espec
o U(k) h ough i e a ions will esul in a mono onous as-
cen sequence φΩ(U(0))≤φΩ(U(1))≤... ≤φΩ(U(k))
ha maximizes φΩ(·). Howe e , his p ope y by i sel does
no gua an ee he con e gence o an ex ac ion solu ion be-
cause decep i e local maxima o φΩ(U(k))migh exis . The
ollowing heo em (whose p oo is ske ched in appendix B)
shows ha his is no he case.
Theo em 2 Unde assump ions A1-A4, he only local ma-
xima o φΩ(U(k))co espond wi h solu ions ha ex ac
one o he independen componen s o he mix u e.
4. EXTRACTION OF SEVERAL SOURCES
In he case o he ex ac ion o Pindependen componen s
(1≤P≤N) he P×Nma ix U(k)is semi-uni a y wi h
ows U(k)
i:, i = 1,...,P. A esul p o ed in [7] s a es
ha any non-nega i e con as ψΩ(y[1],...,y[q])designed
o he ex ac ion o a single sou ce, ha sa is ies (2) and
(20), can be used o cons uc a con as unc ion o he ex-
ac ion o he independen componen s s1( ),··· , sP( ).
Pa icula ized o ou case, his leads o he sequen ial op i-
miza ion, o
ΦΩ(U(k)) =
P
X
i=1
φΩ(U(k)
i:)s. . U(k)(U(k))H=IP.(11)
In able 1 we conside wo choices o he op imiza ion o
he p e ious unc ion which lead o he hin ICA algo i hms
(TICA).
4.1. Hie a chical ex ac ion
A i s op ion (see s ep 5b, 1s choice) is o hie a chically
maximize (11) wi h espec o he ows U(k)
i:, i = 1,...,P
in such a way ha each i h ow sa is ies he cons ain s
U(k)
i:(U(k)
j:)H=δij ∀j≤i, i.e., he i s ows a e less con-
s ained han he las ones. Using Householde e lec ions
he new upda e is exp essed in a e y compac and simple
o m, i is U(k)=QHwhe e Qis a all semi-uni a y ma-
ix o dimension N×Pwhich esul s om he hin QR
decomposi ion2o he weigh ed s a is ic C(k−1)
zy de ined in
s ep 5a o able 1.
4.2. Simul aneous ex ac ion
A second op ion (see s ep 5b, 2nd choice) is o simul ane-
ously maximize ΦΩ(U(k))wi h espec o all he ows o
U(k). A e de ining he He mi ian ma ix o mul iplie s Λ,
he g adien o he Lag angian unc ion associa ed o (11) is
∇LΩ(U(k)) = ∇ΦΩ(U(k))−ΛU(k)(12)
No ing ha ∇ΦΩ(U(k))weakly depends wi h U(k), only
h ough a se o he diagonal e ms, and app oxima ing hese
diagonal e ms by hei cu en es ima es D(k−1)
y(τ)we ob-
ain ∇ΦΩ(U(k))≈(C(k−1)
zy )H. The solu ion o he equa-
ions
(C(k−1)
zy )H=ΛU(k)(13)
U(k)(U(k))H=IP(14)
ΛH=Λ(15)
2The hin QR decomposi ion and he hin Singula Value Decomposi-
ion bo h ha e, o P << N, a compu a ional complexi y o O(NP 2)
lops. An e icien implemen a ion o hem can be ound unde he Ma Lab
commands q (·,0) and s d(·,0).
465
Table 1. Summa y o he hin ICA algo i hms.
1. Se P≤N he numbe o independen
componen s o ex ac om x( ).
2. P ewhi ening z( ) = Wx( )
3. Ini ializa ion
U(0) =IP×N;y(0)( ) = U(0)z( ); k= 1;
4. Es ima e ∀τ∈Ω he ma ices
C(k−1)
zy (τ) = [c(k−1)
zy1(τ),...,c(k−1)
zyq(τ)]
whe e c(k−1)
zyi(τ)is de ined in (18)
and ini ialize U(0) =U(k−1)
5. FOR l= 1 : L
(a) Compu e he diagonal ma ices
D(l−1)
y(τ) = diag U(l−1)C(k−1)
zy (τ)
and he weigh ed sum:
C(l−1)
zy =X
τ∈Ω
wτC(k−1)
zy (τ)(D(l−1)
y(τ))∗
(b) Choice 1. OR 2.

















1. Hie a chical app oach:
[Q,R] = q (C(l−1)
zy ,0)
U(l)=QH
2. Simul aneous app oach:
[Q,∆P×P,V] = s d(C(l−1)
zy ,0)
U(l)=Vsign(∆P×P)QH
END
6. Upda e U(k)=U(L)and es ima e
Pindependen componen s:
y(k)( ) = U(k)z( )
7. IF Con e gence STOP
ELSE k=k+ 1; RETURN TO 4
ha maximizes ΦΩ(U(k))is
[Q,∆P×P,V] = s d(C(k−1)
zy ,0)
U(k)=Vsign(∆P×P)QH(16)
whe e s d(·,0) is he Ma Lab command o he hin singula
alue decomposi ion. The alidi y o he app oxima ion can
be u he imp o ed a i e a ion k, by conside ing Lsubi e -
a ions (l= 1,...,L) o a zigzag p ocedu e which upda es
he es ima es D(l−1)
y(τ)o he diagonal e ms as he solu-
ion changes. This is done in s ep 5. o able 1.
4.3. P ojec ion on o he symme ic subspace
F om heo em 1 one a p io i knows ha he solu ions U[m],
m= 1,...,q which ex ac he independen componen s
belong o he symme ic subspace U[1] =...=U[q]. This
in o ma ion can be exploi ed o imp o e he con e gence o
TICA algo i hms jus adding, a he end o each i e a ion, a
p ojec ion s ep o U(k),...,U(k−q+1) on o his subspace
U(k),U(k−1),...,U(k−q+1) →U(k),...,U(k)(17)
This is ob ained using de ini ion (b) ins ead o (a)
c(k−1)
zyi(τ) =











a) Wi hou p ojec ion:
Cum(z( 1), y(k−1)
i( 2),··· , y(k−q+1)
i( q))
b) Wi h p ojec ion:
Cum(z( 1), y(k−1)
i( 2),··· , y(k−1)
i( q))
(18)
5. SIMULATIONS
In his sec ion we illus a e how he algo i hm (in simila ly
wi h [9]) can be applied o ob ain accu a e es ima es om
educed se o obse a ions. In ou example an a ay o 20
senso s egis e s 250 snapsho s o he obse a ions. These
a e a andom ins an aneous mix u e o 10 independen sig-
nals, in p esence o whi e addi i e Gaussian noise, and wi h
a maximum signal o noise a io o 15dB. The desi ed inde-
penden componen s a e he h ee co ela ed signals ha can
be ob ained, a e no maliza ion, om he il e ing o h ee
bina y p ocesses by he co esponding sys ems F1(z−1) =
(1+0.4z−1+0.9z−2+.5z−3)−1,F2(z−1) = (1+0.6z−1−
0.3z−2)−1and F3(z−1) = (1 −0.7z−1)−1. The o he
se en independen componen s a e samples o empo ally
i.i.d. uni o m p ocesses. We chose second o de s a is ics
q= 2 because o sho da a eco ds like his hey a e usu-
ally he mos eliable, and we se Ω = {( 1, 1−1),( 1, 1−
2),··· ,( 1, 1−7)}because hese se en pai s gua an ee
ha he conside ed independen componen s can be o de ed
acco ding o (2). We un he algo i hm in one hund ed an-
dom expe imen s (wi h di e en andom ma ices, sou ces
and noise samples). In each expe imen we applied he si-
mul aneous TICA algo i hm, wi h P= 3 and L= 2, which
ex ac ed in all he cases he desi ed subse sou ces. As can
be obse ed in igu e 2 he con e gence o he TICA algo-
i hm is qui e as , equi ing be ween 3and 6i e a ions. Un-
o una ely, no p ope compa ison can be gi en wi h o he
algo i hms because SOBI and JADE canno ex ac a subse
o signals while Fas -ICA did no pe o m well o such a
small da a se .
6. DISCUSSION
Recen ly, we no ice ha he con as unc ion p oposed in
heo em 1 admi s a leas squa es cumulan ma ching in e -
466
p e a ion associa ed wi h he ank one app oxima ion o a
se o cumulan enso s. A closely ela ed con as and an
al e na ing Leas Squa es echnique simila o ha we use
we e p e iously p oposed in [11] o sol e he Blind Sou ce
Sepa a ion p oblem. The esul s o his pape b ing a com-
plemen a y insigh o he simul aneous ex ac ion p oblem
and lead o he p oposal o he hin ICA algo i hms. In se-
quel, we commen some o he in e es ing links o hese al-
go i hms wi h o he exis ing app oaches:
•Fo P= 1,q= 4 and Ω = {( , , , )} he TICA al-
go i hms wi h p ojec ion ex ac a single independen
componen by maximizing he modulo o he ku o-
sis o he ou pu . In his case, bo h TICA algo i hms
pa icula ize o he ixed poin algo i hm (Fas -ICA
wi h cubic non-linea i y) p oposed in [5]. Fo a bi-
a y P≤N he hin ICA implemen a ion based
on he hin QR decomposi ion is equi alen o he
hie a chical applica ion o he ixed poin algo i hm
wi h de la ion. Simila ly, he hin ICA implemen a-
ion based on he SVD educes o he ixed poin al-
go i hm wi h symme ic o hogonaliza ion when P=
N(BSS), and p o ides a no el ex ension o i o
P < N (BSE).
•Fo P=Nand q= 2 (al e na i ely q= 4) he
TICA algo i hms ex ac all he independen compo-
nen s using second o de s a is ics ( ou h o de s a is-
ics) and, when i is possible, using also any non-
s a iona i y o he independen componen s. The c i-
e ion (7) is equi alen o ha o he SOBI [9] (JADE
[8]) algo i hm based on he join app oxima e diago-
naliza ion o a ce ain se o cumulan slices, al hough,
he implemen a ion di e s.
•Fo 1< P < N and a bi a y q he TICA algo-
i hms ex ac Pindependen componen s using he
join op imiza ion c i e ia (11). In his case, none o
he p e iously ci ed algo i hms can sol e his p ob-
lem: ixed poin algo i hms (Fas -ICA) does no pe -
o m a join op imiza ion o se e al s a is ics, while
SOBI and JADE implemen a ions a e no sui able o
ex ac ion because he ex ended Jacobi plane o a ions
hey use a e no he mos adequa e echnique o he
es ima ion o a subse o eigen ec o s.
7. CONCLUSIONS
We ha e p oposed a mul i a ia e con as unc ion o he
ex ac ion o a subse o desi ed independen componen s
om a linea mix u e. This con as unc ion join ly op i-
mizes se e al s a is ics o he same o de and ha e no spu-
ious maxima. We ha e sugges ed he hin ICA algo i hms
o he maximiza ion o he con as because hey combine,
0 5 10 15 20 25
10−4
10−2
100
ITERATIONS
PERFORMANCE INDEX
0 1 2 3 4 5 6 7 8 9 10 11
0
0.5
1
1.5
SAMPLE EXPERIMENT
COEFFICIENTS OF |G|
Fig. 2. Uppe ig.: pe o mance index Pindex(G) =
(PN)−1PP
i=1 kGi:k2
2/kGi:k2
∞−1 e sus i e a ions.
Con inuous line is he median cu e o con e gence o 100
expe imen s, he dashed lines deno e he 5 h and 95 h pe -
cen iles. Lowe igu e p esen s he coe icien s o he di e -
en ows o a 3×10 ma ix G o one sample expe imen .
a he same ime, se e al o he ad an ages o o he powe ul
echniques like Fas -ICA, JADE and SOBI.
A. PROOF OF THEOREM 1
The p oo s a s obse ing ha unde assump ions A1-A4
he con as (7) is heo e ically una ec ed by he addi i e
noise. Then, using Cauchy-Schwa z’s inequali y one ob-
ains ha each squa e cumulan wi hin he summa ion is up-
pe bounded by


Cum(y[1]( 1),··· , y[q]( −τq))


2
≤
N
X
j=1
|G[p]
1j|2|Cum(sj( 1),··· , sj( q))|2
·

N
X
j=1 Y
m6=p
|G[m]
1j|2

Since he global ans e ec o s a e no malized
N
X
j=1
q
Y
m6=p
|G[m]
1j|2≤
q
Y
m6=p
N
X
j=1
|G[m]
1j|2= 1 (19)
he e o e,


Cum(y[1]( 1),··· , y[q]( q))


2≤
N
X
j=1
|G[p]
1j|2
·|Cum(sj( 1),··· , sj( q)|2
467

Subs i u ing hese e ms in (7) esul s in
ψΩ(¯y)≤
N
X
j=1
|G[p]
1j|2ψ(sj)p= 1,...,q. (20)
Bu om he o de ing ψΩ(s1)> ψΩ(sj)∀j= 2,...,N in
A4, and he cons ain kG[p]k2= 1 we inally ob ain
ψΩ(y[1],...,y[q])≤ψΩ(s1),(21)
which means ha he uppe bound coincides wi h he ex-
ac ion o he desi ed independen componen . No ing ha
he equali y be ween bo h sides o (19) only holds when he
ow ec o s a e equal G[1] =··· =G[q]and aligned wi h
one o he axis ejT, j = 1,··· , N, one can conclude ha
he global maximum o ψΩ(¯y)is only a ained a he ex ac-
ion o he desi ed sou ce, i.e., when U[1] =··· =U[q]=
(WAe1)Hwhe e e1= (1,0,...,0)T.
B. PROOF OF THEOREM 2
Any c i ical poin U0o φΩ(·)has associa ed a global ex-
ac ion sys em G0=U0WA. Following [12] we de ine
he se o indices o which he elemen s o G0a e nonze o
I={m:G0
1m6= 0, m = 1,...,N},(22)
F om he p oo o heo em 1 one can obse e ha any so-
lu ion which ex ac s one o he independen componen s
(Ihas ca dinali y one) is a local maximum o he con as .
Thus, he decep i e local maximum o φΩ(·), i hey exi ,
should co espond wi h G0ha ing a leas wo nonze o ele-
men s, howe e , hese kind o poin s canno be a maximum
o he unc ion φ(·)because we can always ind local pe -
u ba ion o hem o which he unc ion inc eases. Fo a
local pe u ba ion α∈RNwhe e: kαksu icien small,
PN
j=1 αj= 0,αj= 0 ⇔j6∈ I, and such ha
|G1j|2=|G0
1j|2+αj, j = 1,...,N;(23)
he unc ion φ(·), a he pe u bed poin U, is w i en as
φ(U) = (1 + γ(α))φ(U0) + β(α) + o(kαk2)
whe e
γ(α) = q(q−2)
4X
j∈I





αj
G0
1j





2
(24)
β(α) = X
τ∈Ω
wτ





q
2X
j∈I
αj
(G0
1j)q
|G0
1j|2csj(τ)





2
(25)
Since γ(α)≥0and β(α)>0 o all q≥2we ha e ha
φ(U)> φ(U0),(26)
and we conclude ha any solu ion o which Ihas ca dina-
li y g ea e han 1 canno be a local maximum o φΩ(·).
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