Knowledge-Based Sys ems 241 (2022) 108250
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Knowledge-Based Sys ems
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Consensus g aph and spec al ep esen a ion o one-s ep mul i- iew
ke nel based clus e ing
S. El Hajja b, F. Do naika a,b,c,∗, F. Abdallah d,e, N. Ba ena b
aSchool o Compu e and In o ma ion Enginee ing, Henan Uni e si y, Kai eng, China
bUni e si y o he Basque Coun y UPV/EHU, San Sebas ian, Spain
cIKERBASQUE, Basque Founda ion o Science, Bilbao, Spain
dLebanese Uni e si y, Bei u , Lebanon
eLuxembou g Ins i u e o Socio-Economic Resea ch (LISER), Esch-su -Alze e, Luxembou g
a icle in o
A icle his o y:
Recei ed 29 Ap il 2021
Recei ed in e ised o m 17 Janua y 2022
Accep ed 19 Janua y 2022
A ailable online 25 Janua y 2022
Keywo ds:
Mul i- iew clus e ing
One-s ep clus e ing
G aph lea ning
Spec al ep esen a ion
Nonnega i e embedding
Au oma ic weigh ing
Clus e ing algo i hms
abs ac
Recen ly, mul i- iew clus e ing has ecei ed much a en ion in he ields o machine lea ning and
pa e n ecogni ion. Spec al clus e ing o single and mul iple iews has been he common solu ion.
Despi e i s good clus e ing pe o mance, i has a majo limi a ion: i equi es an ex a s ep o clus e ing.
This ex a s ep, which could be he amous k-means clus e ing, depends hea ily on ini ializa ion, which
may a ec he quali y o he clus e ing esul . To o e come his p oblem, a new me hod called Mul i-
iew Clus e ing ia Consensus G aph Lea ning and Nonnega i e Embedding (MVCGE) is p esen ed
in his pape . In he p oposed app oach, he consensus a ini y ma ix (g aph ma ix), consensus
ep esen a ion and clus e index ma ix (nonnega i e embedding) a e lea ned simul aneously in a
uni ied amewo k. Ou p oposed me hod akes as inpu he di e en ke nel ma ices co esponding
o he di e en iews. The p oposed lea ning model in eg a es wo in e es ing cons ain s: (i) he
clus e indices should be as smoo h as possible o e he consensus g aph and (ii) he clus e indices
a e se o be as close as possible o he g aph con olu ion o he consensus ep esen a ion. In his
app oach, no pos -p ocessing such as k-means o spec al o a ion is equi ed. Ou app oach is es ed
wi h eal and syn he ic da ase s. The expe imen s pe o med show ha he p oposed me hod pe o ms
well compa ed o many s a e-o - he-a app oaches.
©2022 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY-NC-ND
license (h p://c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
1. In oduc ion
Clus e ing is one o he mos impo an esea ch opics in
machine lea ning, which aims o g oup samples in o di e en
g oups, called clus e s, wi hou knowing hei labels [1,2]. In he
las decades, many clus e ing app oaches ha e been de eloped.
In pa icula , mul i- iew clus e ing algo i hms ha e been used
and de eloped o ob ain addi ional in o ma ion o imp o e he
inal clus e ing [3–9]. Among hese me hods, Spec al Clus e ing
(SC) [10–13] me hods a e he mos popula app oaches due o
hei well-de ined ma hema ical amewo k and ease o imple-
men a ion. A e cons uc ing he simila i y ma ix be ween he
da a poin s, hese me hods gene a e a nonlinea p ojec ion o
he da a by mapping he da ase in o a space whe e clus e s
can be easily iden i ied (spec al embedding). A majo d awback
o hese me hods is he use o a pos -p ocessing s ep such as
k-means o ob ain he inal clus e ing esul , which can be a -
ec ed by he ini ializa ion o he p esence o ou lie s. Ma ix
∗Co esponding au ho a : Uni e si y o he Basque Coun y UPV/EHU, San
Sebas ian, Spain.
E-mail add ess: [email p o ec ed] (F. Do naika).
ac o iza ion me hods [14,15] can be used o dimensionali y
educ ion. Fo example, he me hod in [15] called In eg a ion
by Ma ix Fac o iza ion (IMF). This me hod gene a es di e en
ep esen a i e clus e ing ma ices compu ed independen ly o
each iew, gene a es an in e media e ma ix o all iews, and
hen pe o ms a ac o iza ion p ocess on his ma ix o econcile
he di e en clus e ing ma ices gene a ed om he di e en
iews. Ma ix ac o iza ion me hods ha e a low compu a ional
cos compa ed o o he me hods. This is because ac o iza ion
o a pa icula ma ix decomposes i in o i s cons i uen pa s,
which can also simpli y he ma ix ope a ions as hey a e applied
o he ob ained ma ices and no o he o iginal complex ma-
ix. Howe e , hese me hods canno deal wi h he nonlinea i y
o he da a. To deal wi h he p oblem o nonlinea i y o da a,
se e al app oaches ha e p oposed a solu ion based on mul iple
ke nels [7,16,17]. Wi h hese me hods, he da a is mapped in o
a space whe e i is linea ly sepa able. Howe e , i is no ed ha
mul i- iew clus e ing s ill needs imp o emen . To add ess his
issue, in his pape we p esen a no el app oach ha p o ides a
consis en non-nega i e embedding ma ix o de e mine he inal
clus e assignmen . Ou p oposed me hod es ima es he clus e -
ing o he da a di ec ly wi hou any addi ional pos -p ocessing.
h ps://doi.o g/10.1016/j.knosys.2022.108250
0950-7051/©2022 The Au ho s. Published by Else ie B.V. This is an open access a icle unde he CC BY-NC-ND license (h p://c ea i ecommons.o g/licenses/by-
nc-nd/4.0/).
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
I en o ces ha he clus e index ma ix is a kind o con olu ion
o a uni ied spec al ep esen a ion o e a consis en g aph. The
me hod we p opose is called mul i- iew clus e ing ia consensus
g aph lea ning and nonnega i e embedding (MVCGE). I can o e -
come some d awbacks o o he app oaches. The p oposed me hod
can simul aneously p o ide he consis en simila i y g aph, he
non-nega i e clus e index ma ix and he uni ied spec al p o-
jec ion ma ix ac oss all iews. Mo eo e , his me hod au o-
ma ically calcula es he weigh o each iew wi hou using any
addi ional pa ame e s. The p oposed me hod combines he ad-
an ages o g aph-based me hods and mul iple ke nel me hods.
In o he wo ds, ou me hod e ains wo in e es ing p ope ies
ha he cu en me hods NESE in [18] and MVCSK in [19] do
no ha e simul aneously. The i s p ope y, inspi ed by NESE, is
he non-dependence on a pa icula clus e ing algo i hm such as
k-means clus e ing. The o he p ope y (inspi ed by MVCSK) is
he simul aneous es ima ion o a consis en uni ied g aph and a
uni ied spec al ep esen a ion. Since ou me hod combines he
ad an ages o NESE and MVCSK, he main goal o ou s udy is
o ou pe o m he p e ious wo me hods. The e o e, hese wo
me hods a e used as he main compe ing me hods in ou s udy.
The con ibu ions o he pape a e summa ized below.
1. Unlike o he app oaches based on mul ile el lea ning, ou
me hod can simul aneously p o ide he consensus sim-
ila i y ma ix, he nonnega i e index clus e ma ix, he
spec al p ojec ion ma ix, and he weigh o each iew
au oma ically.
2. I gene a es he inal clus e ing assignmen di ec ly wi h-
ou any pos -p ocessing s ep. Ou me hod inhe i s he ad-
an ages o ma ix ac o iza ion me hods and g aph based
me hods.
3. The p oposed model success ully inds nonlinea in e -
ac ions be ween di e en iews. This me hod is able o
compu e he exac g aph conside ing he unde lying co e-
la ions om nume ous iews by using a ke nel ep esen-
a ion o each iew.
4. The clus e index ma ix, which is he consequence o he
con olu ion o he cohe en spec al p ojec ion ma ix o e
he cohe en g aph, is lea ned as pa o he p oposed
lea ning echnique.
5. I has been alida ed on eal and syn he ic da ase s. This
alida ion shows ha his app oach can gi e be e esul s
compa ed o s a e-o - he-a clus e ing me hods.
The es o he pape is o ganized as ollows. Sec ion 2in oduces
he main concep s and some ela ed wo k. Sec ion 3desc ibes ou
p oposed me hod in de ail. Sec ion 4 epo s expe imen al esul s
ob ained on eal and syn he ic da ase s. Sec ion 5concludes he
pape .
2. P elimina ies and ela ed wo k
2.1. No a ions
We conside a da a ma ix X wi h nda a poin s as (x
1,x
2,...,
x
n)∈Rd ×n, whe e d is he numbe o ea u es in he co e-
sponding iew . We ep esen he ma ices in bold uppe case
le e s, he ec o s in bold lowe case le e s, and he cons an s in
non-bold le e s. The ace o a ma ix Mis deno ed by T (M) and
i s anspose by MT. The F obenius no m o Mis gi en by ∥M∥F=
∥M∥2=√∑n
i=1∑d
j=1|Mij|2.I,1n,Dand La e he iden i y ma ix,
he column ec o wi h nelemen s equal o one, he diagonal
ma ix, and he Laplacian ma ix o he g aph, espec i ely. The
numbe o clus e s is deno ed by K. The simila i y ma ix, he
nonnega i e embedding ma ix, he spec al p ojec ion ma ix,
and he ke nel ma ix a e deno ed by S,H,Pand K, espec i ely.
2.2. Rela ed wo k
Recen ly, se e al mul i iew clus e ing app oaches ha e been
p oposed. The cu en app oaches can be di ided in o se e al
g oups: Spec al clus e ing algo i hms [12,13,20,21], G aph based
clus e ing algo i hms [22], Weigh ed mul i iew clus e ing ap-
p oaches [10,21,23–25], Au oma ically weigh ed mul i iew
clus e ing algo i hms [19,26–28], Mul i iew subspace based clus-
e ing app oaches [29,30], Ke nel based App oaches [19], Ma ix
ac o iza ion app oaches [15,18,31], Nonnega i e ma ix ac o -
iza ion me hods [18,32], e c. In his sec ion, we p esen se e al
me hods ha belong o hese ca ego ies. A popula ca ego y
o app oaches is Spec al clus e ing [12,13,20,21]. This me hod
cons uc s a simila i y g aph be ween da a poin s and hen con-
s uc s a da a ep esen a ion ma ix using he eigen ec o s o
he co esponding Laplacian o he g aph. The e o e, a pos -
p ocessing s ep is used o ob ain he inal clus e ing assignmen .
An example o a spec al clus e ing algo i hm is he amous co-
aining clus e ing algo i hm [12], which adjus s he simila i y
ma ix o a gi en iew based on he clus e ing esul o ano he
iew so ha he same ins ance is placed in he same clus e in
di e en iews.
Mo eo e , co- egula ed spec al clus e ing [13] combines he
simila i y ma ices ob ained om di e en iews wi h an adap-
i e scheme o ob ain he esul . weigh ed mul i iew clus e ing
app oaches [10,23–25] assign weigh s o each iew o accoun
o he con ibu ion o each iew o he inal clus e ing esul .
Mo eo e , hese app oaches use a consis en scheme o me ge
he di e en iews. A d awback o hese app oaches is he use
o addi ional weigh ing pa ame e s. To o e come his d awback,
Au oma ically weigh ed mul i- iew clus e ing algo i hms ha e
been p oposed [19,26–28]. Mo eo e , ano he app oach ela ed
o he abo e ca ego ies is p esen ed, which ex ends he spec-
al clus e ing algo i hm wi h he idea o weigh ed iews. This
me hod is called Adap i e Weigh ed P oc us es (AWP) [21]. The
inal clus e ing assignmen o his me hod is achie ed by spec al
o a ion. Mo eo e , his me hod p o ides accu a e clus e ing wi h
low compu a ional cos . Ano he amous ca ego y named Mul i-
iew Subspace based Clus e ing app oach (MVSC) was in oduced
in [29,30] o lea n he bes and consis en ep esen a ion o
he da a. Mo eo e , a Mul i- iew Lea ning me hod wi h Adap-
i e Neighbo s (MLAN) is p oposed in [33] o join ly lea n he
simila i y g aph and pe o m he inal clus e ing assignmen .
The ke nel-based app oaches (e.g., [19]) a e used o o e come
he p oblem o nonlinea i y o he da a by mapping hem o a
space in which hey a e linea ly sepa able, and hen hey sol e
he p oblem caused by he mul iple shapes o he da a. The ma ix
ac o iza ion app oaches (e.g., [15,18,31]) a e used because o
hei low compu a ional cos , which makes hem e icien o
dimensionali y educ ion. They can p o ide high clus e ing pe -
o mance compa ed o o he me hods. Howe e , his ca ego y o
clus e ing me hods canno handle nonlinea da a.
In [32], he au ho s p opose a Non-nega i e Ma ix Fac o iza-
ion (NMF) app oach ha uses dual cons ain s. This app oach
exploi s he labeling o some images and he spa si y o ep e-
sen a ions. In [34], he au ho s in oduced a uni ied amewo k
o Join clus e ing and dis ance me ic lea ning. These a e sol ed
ia ank educed eg ession. The app oach p o ided some new
insigh s o lea ning a clus e ing ha adop s dis ance me ic
lea ning. In [35], he au ho s p esen ed an Ensemble clus e ing
me hod based on e icien p opaga ion o clus e wise simila i ies
ia andom walks. The wo k o [36] p oposed a Locali y Adap i e
La en Mul iView Clus e ing (LALMVC) me hod. I simul aneously
lea ns he la en consensus ep esen a ion ia linea ans o ma-
ions, he join spec al ep esen a ion and he consensus g aph.
The lea ned consensus g aph ma ix is hen used in spec al
clus e ing o ob ain a clus e index ma ix.
2
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
Fig. 1. Illus a ion o he p oposed me hod.
In [37], he au ho s p opose a model in which he indi idual
g aphs, a used g aph, and a spec al p ojec ion a e es ima ed
simul aneously. Sel - ep esen a i eness o he da a was used in
es ima ing he indi idual g aphs. In [38], a non-nega i e ma-
ix ac o iza ion wi h mul iple iews is p oposed. The model
es ima ing he iew-based wo non-nega i e ma ices in eg a es
mani old egula iza ion in he low-dimensional subspace and
he pai wise consis encies o in e iew simila i y in hese low-
dimensional subspaces. In [17], he au ho s join ly es ima e an
op imal g aph and an adequa e consensus ke nel o clus e ing
by o cing he global ke nel ma ix o be a con ex combina-
ion o a se o basis ke nels. Thei p oposed model en o ces a
egula iza ion o he uni ied g aph and he inal ke nel ma ix.
In [39], he au ho s use he co en opy-induced me ic (CIM)
o deal wi h he noise ha exis s in each iew. They use iew-
speci ic embedding om an in o ma ion heo e ic pe spec i e.
In [40], he au ho s p opose he algo i hm C oss- iew Ma ching
Clus e ing (COMIC), which can clus e da a wi h mul iple iews.
The algo i hm can also es ima e he numbe o clus e s. COMIC
p o ides c oss- iew consensus on iew-speci ic simila i y g aphs
ins ead o iew-speci ic da a ep esen a ions.
In [41], he au ho s p o ide an o e iew o mul i- iew clus-
e ing. This su ey desc ibes a wide ange o mul i- iew clus-
e ing me hods, including bo h gene a i e and disc imina i e
app oaches. Fu he mo e, he au ho s o his su ey di ide hese
algo i hms in o many g oups and gi e nume ous examples o
how hey a e used o mul i- iew clus e ing. In [42], he au-
ho s in oduce a me hod called Mul i- iew clus e analysis wi h
incomple e da a o unde s and ea men e ec s. Indeed, some-
imes da a en ies a e missing in se e al o he iews. Cu en
mul i iew co-clus e ing app oaches a e no able o success ully
deal wi h incomple e da a, especially when he e a e many pa -
e ns o incomple e da a. By using an indica o ma ix whose
en ies indica e which da a i ems a e p esen , and measu ing
clus e ing pe o mance based solely on he obse ed alues,
his me hod p o ides an imp o ed app oach o mul i- iew co-
clus e ing algo i hms o deal wi h he missing da a p oblem.
Mo eo e , his me hod is less p one o impu a ion unce ain y
han s anda d me hods ha subs i u e missing da a o pe o m
egula mul i- iew da a clus e ing.
3. P oposed app oach
We in oduce a new app oach called Mul i-View Clus e -
ing ia Consensus G aph Lea ning and Nonnega i e Embedding
(MVCGE), which combines he ad an ages o g aph lea ning
me hods and ma ix ac o iza ion me hods. MVCGE achie es he
clus e ing esul s wi hou any addi ional s ep. Fig. 1 shows an
illus a ion o ou p oposed mul i- iew clus e ing me hod.
The p oposed me hod can simul aneously es ima e (1) he
consensus simila i y ma ix, (2) he consensus da a ep esen-
a ion ma ix, and (3) he nonnega i e clus e index ma ix.
Mo eo e , he weigh o each iew is au oma ically upda ed
wi hou any addi ional pa ame e s. Gi en nsamples and V iews
( ea u e ec o s), he da a ma ix o each iew can be ep e-
sen ed as X = [x
1,x
2,...,x
n] ∈ Rd ×n, whe e d ep esen s
he numbe o ea u es in he co esponding iew, whe e =
1,...,V. The co esponding ke nel ma ices a e deno ed by K .
The da ase is o be g ouped in o Kclus e s based on he V
iews. The unknown ma ices a e S∈Rn×n,P∈Rn×K, and
H∈Rn×K. Ou p oposed me hod es ima es hese ma ices
simul aneously by in eg a ing se e al p ope ies such as g aph
cons uc ion using sel - ep esen a ion o da a, smoo hness o
clus e labels, and spec al da a con olu ion. Thus, ou p oposed
c i e ion has h ee main e ms. To ob ain he i s e m o ou
p oposed c i e ion, we used he idea o MVCSK me hod in [19].
To es ima e a consis en g aph ma ix, his me hod exploi s
he p ope y o he da a o exp ess i sel , whe e he da a is
mapped nonlinea ly. The e o e, he consis en g aph ma ix S
should sa is y he condi ion min ∑V
=1∥Φ(X )−Φ(X )S∥ =
∑V
=1√T (K −2K S+STK S), whe e K =Φ(X )TΦ(X )
and Φ() is a gi en nonlinea mapping, which should no be
explici ly s a ed, since only he knowledge o he ke nel ma ix K
is needed. Mo eo e , o a oid he i ial solu ion o he consis en
g aph ma ix, a egula iza ion e m is used o con ol he alues
in his ma ix. The i s e m is as ollows:
min
S
V
∑
=1√T (K −2K S+STK S)+α∥S∥2
2s. .S≥0.(1)
3
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
Fig. 2. Visualiza ion o he o iginal syn he ic da ase s: (a) Te a, (b) Hep a, and (c) Chainlink.
I is also impo an o assign a weigh pa ame e o each iew o
ep esen he con ibu ion o each iew o he clus e ing p ocess.
The squa e oo in Eq. (1) is used o au oma ically upda e he
weigh o each iew [19] au oma ically. The weigh o he iew
w is gi en by:
w =1
2√T (K −2K S+STK S)
.(2)
By using he weigh exp ession in Eq. (2), i can be shown ha
p oblem (1) is equi alen o he ollowing p oblem:
min
S
V
∑
=1
w T (K −2K S+STK S)+α∥S∥2
2s. .S≥0.(3)
The clus e ing esul is ob ained om he nonnega i e em-
bedding ma ix H, which p o ides he clus e indices by aking
he index o he highes elemen in he ow ec o Hi∗∈RK.
Since he ma ix His used o he inal clus e assignmen , i
is impo an o use a smoo hing e m o his ma ix so ha i
is mo e cohe en wi h he g aph en ies. The smoo hing e m
ensu es ha wo da a poin s x
iand x
j ha a e simila (i.e., he
alue o he co esponding alue in he simila i y ma ix Sij is
la ge) a e necessa ily in he same clus e (i.e., he co esponding
clus e index Hi∗and Hi∗a e close). The e o e, he second e m
o ou c i e ion is gi en by:
min
H
1
2∑
i∑
j
∥Hi∗−Hj∗∥2Sij =min
HT (HTL H),(4)
whe e L=D−S∈Rn×nis he Laplacian ma ix o he consis en
g aph ma ix, and Dis a diagonal ma ix whose elemen s a e
gi en by: Dii =∑n
j=1
Sij+Sji
2. The hi d e m o ou p oposed
me hod s a es ha he clus e index o he i h ins ance ( he ow
ec o Hi,∗) is se o he con olu ion o he spec al ep esen a ion
Pwi h he i h ow o he g aph ma ix Si,∗. This app oach has
wo main ad an ages. Fi s , he clus e ing is pe o med in a
single s ep. Second, he clus e ing uses he consolida ed spec al
ep esen a ion o he neighbo s ob ained in he consensus g aph.
Mo eo e , inspi ed by he p inciple o da a con olu ion, he
nonnega i e embedding ma ix used o ob ain he inal clus e ing
assignmen will be equal o ‘‘H=max(S P,0)". This means ha
he nonnega i e ma ix is he esul o he con olu ion o he
spec al da a ep esen a ion wi h he g aph. The hi d e m o ou
c i e ion binds he clus e index label o he consensus spec al
ep esen a ion. The e o e, he clus e index ma ix should sa is y
he ollowing condi ion:
min
H≥0∥H−S P∥2
F(5)
whe e he ma ix P∈Rn×Kis a consensus da a ep esen a ion.
In ou wo k, i is ini ialized o a uni ied spec al ep esen a-
ion o he da a. The p oposed me hod unes his ep esen a ion
acco ding o a global objec i e.
Since he ma ix Pis o hogonal, Eq. (5) can ake ano he
o m o illus a e he ac o iza ion o he g aph ma ix Susing
he nonnega i e embedding ma ix Hand he spec al p ojec ion
ma ix P. I can be w i en as:
min
H,P∥S−H PT∥2
Fs. .H≥0,PTP=I.(6)
Ou inal objec i e unc ion is ob ained by adding he h ee
e ms om Eqs. (1),(4), and (6).
min
S,P,H
V
∑
=1
w T (K −2K S+STK S)+α∥S∥2
2
+λ1T (HTL H)+λ2∥S−H PT∥2
2
s. .S≥0,PTP=I,HTH=I,H≥0,(7)
whe e α,λ1and λ2a e h ee egula iza ion pa ame e s.
Op imiza ion. We use an i e a i e upda e p ocedu e o sol e
ou objec i e unc ion. In MVCGE, h ee ma ices a e unknown:
S,H, and P. An al e na ing op imiza ion scheme is used o he
op imiza ion p ocedu e. We p oceed as ollows:
S ep 1: Fix all, es ima e H: The p oblem (7) is:
min
HT (HTL H)+λ2
λ1
∥S P −H∥2
2s. .HTH=I,H≥0.(8)
Vanishing he de i a i e o (8) w. . . Hyields:
H=(L+λ2
λ1
I)−1λ2
λ1
S P.(9)
To sa is y he o hogonali y and non-nega i i y cons ain s, an
o hogonaliza ion s ep is i s applied o he ob ained H, hen he
nega i e alues o Ha e se o ze o.
S ep 2: Fix all, es ima e P: The p oblem (7) becomes:
min
P∥S−H PT∥22.(10)
Since Pis o hogonal, i.e., PTP=I,Pis ob ained by pe o ming
he singula alue decomposi ion o STH. Le UΣVT=SVD (STH),
hen he solu ion o (10) is gi en by:
P=U VTwi h UΣVT=SVD (STH).(11)
S ep 3: Fix all, es ima e S:
4
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
Fig. 3. -SNE o he spec al p ojec ion and nonnega i e embedding ma ices ob ained by he p oposed clus e ing me hod MVCGE o di e en da ase s.
I we ix Hand P, we need o sol e he ollowing p oblem:
min
S
V
∑
=1
w T (K −2K S+STK S)+α∥S∥2
2
+λ1T (HTL H)+λ2∥S−H PT∥2
2s. .S≥0.(12)
A e he spec al clus e ing analysis, we ha e he known iden-
i y:
T (HTL H)=1
2∑
i∑
j
Hi∗−Hj∗
2Sij =T (Q S),(13)
whe e Hi∗is he i h ow o H. The symme ic ma ix Qdeno es he
pai wise dis ance associa ed wi h he ows o he ma ix H. I is
gi en by Qij =1
2
Hi∗−Hj∗
2. Subs i u ing Eq. (13) in o Eq. (12),
he la e becomes:
min
S
V
∑
=1
w T (K −2K S+STK S)+α∥S∥2
2
+λ1T (Q S)+λ2∥S−H PT∥2
2s. .S≥0.(14)
5
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
Fig. 4. Visualiza ion o he wo clus e s ob ained by h ee di e en me hods o he Chainlink da ase .
By making he de i a i e o Eq. (14) w. . . S anish, we ob ain S
as (ReLU() is he Rec i ied Linea Uni unc ion):
S=ReLU ⎧
⎨
⎩(V
∑
=1
w K +(α+λ2)I)−1
×(V
∑
=1
w K +λ2H PT−1
2λ1Q)}.(15)
S ep 4: Fix H,P,and S,and upda e w ( =1,...,V)using
Eq. (2).
The main s eps o he p oposed app oach ‘‘Mul i- iew Clus e -
ing ia Consensus G aph Lea ning and Nonnega i e Embedding"
(MVCGE) a e summa ized in Algo i hm 1.
Algo i hm 1 MVCGE
Inpu : Da a samples in V iews X ∈Rn×d , =1,...,V.
The g aph ma ices S , =1,...,V.
The spec al embedding ma ices P , =1,...,V.
Pa ame e s α,λ1,λ2.
Ou pu : The consensus g aph ma ix S.
The consensus spec al ep esen a ion ma ix P.
The clus e index ma ix (nonnega i e embedding
ma ix) H.
Ini ializa ion:
The weigh o each iew w =1
V.
Compu e he ke nel ma ix K o each iew.
Ini ialize Sand Pby aking he a e age o he
ma ices S and P .
Repea
Upda e Husing Eq. (9).
Upda e Pusing Eq. (11).
Upda e Susing Eq. (15).
Upda e w using Eq. (2).
Un il con e gence
To ini ialize he wo ma ices Sand P, he e icien me hod
used in [43] is used. This me hod inds he simila i y ma ix and
he co esponding spec al p ojec ion ma ix o each iew. To
6
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
Table 1
Desc ip ion o he eal da ase s used in he pape .
View COIL20 ORL Ou -Scene BBCSpo MSRC 1
1In ensi y-1024 GIST-512 GIST-512 In ensi y-3183 GIST-512
2LBP-3304 LBP-59 LBP-48 LBP-3203 LBP-256
3Gabo -6750 HOG-864 HOG-256 – Colo momen -24
4– Cen is -254 Colo mom.-432 – Cen is -254
5– – – – Si -512
# Samples 1440 400 2688 544 210
# Classes 20 40 8 5 7
View Ex ended-Yale MNIST MNIST-1000
1Co a iance ch9 g ay-45 Resne 50-2048 Resne 50 Pooling-2048
2LBP-900 VGG16-4096 VGG16 FC1-4096
3– – –
4– – –
5– – –
# Samples 1774 10000 1000
# Classes 28 10 10
ob ain he ini ial uni ied ma ix, he a e age o all he indi idual
ma ices is used.
4. Pe o mance e alua ion
4.1. Da ase s
The e ec i eness o he p oposed app oach is e alua ed using
eigh eal image da ase s and h ee syn he ic da ase s. The MNIST
da ase is ela i ely la ge. Table 1 desc ibes he eal da ase s.
We also used h ee syn he ic da ase s: Te a, Hep a, and Chain-
link. They we e selec ed om he Fundamen al Clus e ing P ob-
lem Sui e (FCPS). Fo hese da ase s, only one iew is conside ed.
Te a con ains 400 3D poin s di ided in o ou g oups. Hep a
con ains 212 3D poin s g ouped in o se en well-de ined clus e s
wi h di e en a iances. Chainlink is o med by wo clus e s ha
a e no linea ly sepa able. I consis s o 1000 3D poin s. These
da ase s a e isualized in Fig. 2 [44]. All hese syn he ic da ase s
use 3D da a poin s pi∈R3. The 3-dimensional da ase s a e
ans o med in o high-dimensional da ase s xi∈R100 using he
ollowing linea and nonlinea mappings xi=σ(Uσ(W pi))
whe e he sigmoid unc ion σis used o in oduce nonlinea i y,
W∈R10×3and U∈R100×10 a e wo ma ices whose en ies
ollow he Gaussian dis ibu ion wi h ze o-mean uni a iance
i.i.d.
4.2. Expe imen al se up
Se e al compe ing me hods a e used o compa ison: (1) Co-
aining app oach o mul i- iew Spec al Clus e ing (Co SC) [12],
(2) Co- egula ized app oach o mul i- iew Spec al Clus e ing
(Co SC) [13], (3) Mul i- iew Lea ning Clus e ing wi h Adap i e
Neighbo s (MLAN) [33], (4) Sel -weigh ed Mul i- iew Clus e ing
wi h mul iple g aphs (SwMC) [45], (5) A ini y Agg ega ion o
Spec al Clus e ing (AASC) [46], (6) G aph Lea ning o Mul i-
View clus e ing (MVGL) [22], (7) Pa ame e - ee Au o-weigh ed
Mul iple G aph Lea ning (AMGL) [28], (8) Mul i- iew clus e -
ing ia Adap i ely Weigh ed P oc us es (AWP) [21], (9) Au o-
weigh ed Mul i-View Clus e ing ia Ke nelized g aph lea ning
(MVCSK) [19], (10) Mul i- iew spec al clus e ing ia in eg a ing
Non-nega i e Embedding and Spec al Embedding (NESE) [18],
(11) Spa se Mul i- iew Spec al Clus e ing (S-MVSC) [47], (12)
Consis ency-awa e and Inconsis ency-awa e G aph-based Mul i-
View Clus e ing (CI-GMVC) [48], (13) Mul i-View Clus e ing in La-
en Embedding Space (MCLES) [49] and (14) mul i- iew spec al
clus e ing ia Cons ained Nonnega i e Embedding (CNESE) [9].
We also epo he Spec al Clus e ing bes iew esul (SC) [20].
The clus e ing pe o mance o he p oposed app oach is com-
pa ed wi h o he me hods by using he au ho s’ sou ce codes
wi h he de aul o p oposal pa ame e se ings,1o by di ec ly
epo ing he bes expe imen al esul s om he co esponding
published pape s.2A Gaussian ke nel unc ion is used o cons uc
he ke nel ma ix o each iew. To ini ialize ou algo i hm, we
use he same me hod as in [18], which cons uc s he simila i y
ma ix o each iew acco ding o a smoo hing cons ain , an
ℓ2 egula iza ion e m, and a non-nega i i y cons ain . Then,
he co esponding spec al p ojec ion ma ix o each iew is
compu ed, and he inal uni ied simila i y ma ix and spec al
p ojec ion ma ix is he a e age o he co esponding ma ix o
all iews. In his way, we ob ain he ini ial alues o he ma ices
Sand P.
In ou me hod, h ee pa ame e s a e used: α,λ1and λ2. The
alues o αa e in he ange [0.005 0.9], he alues o he pa am-
e e λ1 a y o e he se {10−10, 10−9, 10−8, 10−7, 10−6, 10−5,
10−4, 10−3} and he alues o he pa ame e λ2 a y o e he se
{10−7, 10−6, 10−5, 10−410−3, 10−2, 10−1}. In ou expe imen s, he
ange o each pa ame e is chosen o encompass a wide ange.
This ensu es ha he op imal alues o hese pa ame e s a e
wi hin his ange. In selec ing he alues in hese anges, we used
a g id sea ch me hod. G id sea ch is a me hod o exhaus i ely
sea ching a manually de ined subse o he pa ame e space o
a gi en algo i hm. The numbe o pa ame e s o he algo i hm
is he spa ial dimension o he g id. So in ou case, he g id is
in a 3D space. This me hod s a s by c ea ing he g id, sampling
he p ede ined egions. Then, o each pa ame e combina ion
( ep esen ed by he nodes o he g id), a model is c ea ed o ind
he bes pa ame e combina ion ha p o ides he bes clus e ing
pe o mance. The bes clus e ing esul is indica ed by a clus e
pe o mance me ic. To compa e ou me hod wi h o he me hods,
we use ou clus e ing pe o mance me ics: Accu acy (ACC),
No malized Mu ual In o ma ion (NMI), Pu i y, and Adjus ed Rand
Index (ARI). Thei de ini ion can be ound in [50].
4.3. Expe imen al esul s
Ou algo i hm is es ed on eal and syn he ic da ase s. Table 2
shows he esul s ob ained by MVCGE and some o he me hods
on he da ase s: ORL, Ou -Scene, and Coil20. In his able, he
highes sco es a e ma ked in bold. The p oposed me hod MVCGE
was supe io on hese da ase s. Fo some compe ing me hods
lis ed in Table 2, he co esponding me hod is epea ed in mul-
iple ials, and hen a s anda d de ia ion o each indica o is
gi en in pa en heses. F om his able, we can see ha ou me hod
1This conce ns SC, MVCSK, NESE, S-MVSC, CI-GMVC, MCLES and CNESE.
2This conce ns he ollowing me hods: Co SC, Co SC, MLAN, SwMC, AASC,
MVGL, AMGL and AWP.
7
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
Table 2
Clus e ing pe o mance on he ORL, Ou doo -Scene and Coil20 da ase s.
Da ase Me hod ACC NMI Pu i y ARI
ORL SC-Bes [20] 0.66 (±0.02) 0.76 (±0.02) 0.71 (±0.02) 0.67 (±0.01)
AWP [21] 0.80 (±0.00) 0.91 (±0.00) 0.83 (±0.00) 0.76 (±0.00)
MLAN [33] 0.78 (±0.00) 0.88 (±0.00) 0.82 (±0.00) 0.67 (±0.00)
SwMC [45] 0.77 (±0.00) 0.90 (±0.00) 0.83 (±0.00) 0.62 (±0.00)
AMGL [28] 0.75 (±0.02) 0.90 (±0.02) 0.82 (±0.02) 0.63 (±0.09)
AASC [46] 0.82 (±0.02) 0.91 (±0.01) 0.85 (±0.01) 0.76 (±0.02)
MVGL [22] 0.75 (±0.00) 0.88 (±0.00) 0.80 (±0.00) 0.55 (±0.00)
Co SC [13] 0.77 (±0.03) 0.90 (±0.01) 0.82 (±0.03) 0.72 (±0.04)
Co SC [12] 0.75 (±0.04) 0.87 (±0.01) 0.78 (±0.03) 0.67 (±0.03)
NESE [18] 0.82 (±0.00) 0.91 (±0.00) 0.85 (±0.00) 0.75 (±0.00)
MVCSK [19] 0.85 (±0.02) 0.94 (±0.01) 0.88 (±0.02) 0.81 (±0.02)
S-MVSC [47] 0.80 (±0.02) 0.93 (±0.01) 0.82 (±0.02) 0.89 (±0.01)
CI-GMVC [48] 0.81 (±0.00) 0.92 (±0.00) 0.85 (±0.00) 0.74 (±0.00)
MCLES [49] 0.84 (±0.00) 0.94 (±0.00) 0.88 (±0.00) 0.79 (±0.00)
CNESE [9] 0.87 (±0.00) 0.95 (±0.00) 0.89 (±0.00) 0.84 (±0.00)
MVCGE 0.93 (±0.00)0.97 (±0.00)0.95 (±0.00)0.92 (±0.00)
Ou -Scene SC-Bes [20] 0.47 (±0.01) 0.39 (±0.01) 0.57 (±0.01) 0.34 (±0.01)
AWP [21] 0.65 (±0.00) 0.51 (±0.00) 0.65 (±0.00) 0.42 (±0.00)
MLAN [33] 0.55 (±0.02) 0.47 (±0.01) 0.55 (±0.02) 0.33 (±0.03)
SwMC [45] 0.50 (±0.00) 0.47 (±0.00) 0.50 (±0.00) 0.38 (±0.00)
AMGL [28] 0.51 (±0.05) 0.45 (±0.03) 0.52 (±0.04) 0.34 (±0.05)
AASC [46] 0.60 (±0.00) 0.48 (±0.00) 0.60 (±0.00) 0.35 (±0.00)
MVGL [22] 0.42 (±0.00) 0.31 (±0.00) 0.43 (±0.00) 0.16 (±0.00)
Co SC [13] 0.51 (±0.04) 0.39 (±0.03) 0.52 (±0.03) 0.31 (±0.02)
Co SC [12] 0.38 (±0.02) 0.22 (±0.01) 0.39 (±0.02) 0.16 (±0.01)
NESE [18] 0.63 (±0.00) 0.53 (±0.00) 0.66 (±0.00) 0.46 (±0.00)
MVCSK [19] 0.65 (±0.01) 0.52 (±0.00) 0.65 (±0.01) 0.42 (±0.00)
S-MVSC [47] 0.48 (±0.01) 0.54 (±0.02) 0.65 (±0.01) 0.46 (±0.04)
CI-GMVC [48] 0.35 (±0.01) 0.31 (±0.00) 0.35 (±0.01) 0.19 (±0.00)
MCLES [49] 0.65 (±0.00) 0.53 (±0.00) 0.67 (±0.00) 0.46 (±0.00)
CNESE [9] 0.66 (±0.00) 0.55 (±0.00) 0.67 (±0.00) 0.47 (±0.00)
MVCGE 0.70 (±0.00)0.55 (±0.00)0.70 (±0.00)0.47 (±0.00)
COIL20 SC-Bes [20] 0.73 (±0.01) 0.82 (±0.01) 0.75 (±0.01) 0.68 (±0.02)
AWP [21] 0.68 (±0.00) 0.87 (±0.00) 0.75 (±0.00) 0.71 (±0.00)
MLAN [33] 0.84 (±0.00) 0.92 (±0.00) 0.88 (±0.00) 0.81 (±0.00)
SwMC [45] 0.86 (±0.00) 0.94 (±0.00) 0.90 (±0.00) 0.84 (±0.00)
AMGL [28] 0.80 (±0.04) 0.91 (±0.02) 0.85 (±0.03) 0.74 (±0.07)
AASC [46] 0.79 (±0.00) 0.89 (±0.00) 0.83 (±0.00) 0.76 (±0.00)
MVGL [22] 0.78 (±0.00) 0.88 (±0.00) 0.81 (±0.00) 0.75 (±0.00)
Co SC [13] 0.68 (±0.04) 0.78 (±0.02) 0.70 (±0.03) 0.62 (±0.03)
Co SC [12] 0.70 (±0.03) 0.80 (±0.02) 0.72 (±0.03) 0.65 (±0.03)
NESE [18] 0.77 (±0.00) 0.88 (±0.00) 0.82 (±0.00) 0.69 (±0.00)
MVCSK [19] 0.65 (±0.04) 0.80 (±0.02) 0.70 (±0.03) 0.61 (±0.05)
S-MVSC [47] 0.62 (±0.01) 0.86 (±0.02) 0.77 (±0.02) 0.97 (±0.02)
CI-GMVC [48] 0.86 (±0.00) 0.94 (±0.00) 0.90 (±0.00) 0.83 (±0.00)
MCLES [49] 0.79 (±0.00) 0.88 (±0.00) 0.83 (±0.00) 0.75 (±0.00)
CNESE [9] 0.82 (±0.00) 0.88 (±0.00) 0.82 (±0.00) 0.78 (±0.00)
MVCGE 1.00 (±0.00)1.00 (±0.00)1.00 (±0.00)1.00 (±0.00)
and me hods MVCSK, NESE S-MVSC, CI-GMVC, MCLES and CNESE
pe o m bes , so we can adop hem o es he o he da ase s.
Table 3 shows a compa ison be ween ou me hod and he
a o emen ioned me hods o he BBCSpo , MSRC 1, Ex ended-
Yale, MNIST and MNIST-1000 da ase s. Fo he MNIST da ase ,
which is a la ge image da ase (i.e., he numbe o samples is
equal o 10000), each image has wo deep desc ip o s, which
means ha he da a al eady has some nonlinea i y. Then, he
use o he la ge ke nel ma ices can be skipped. The e o e, he
c i e ion o ou me hod educes o he las wo e ms, whe e we
only upda e he spec al p ojec ion ma ix and he non-nega i e
embedding ma ix.
Ou me hod is applied o he syn he ic da ase s: Te a, Chain-
link, and Hep a. The esul s a e p esen ed in Table 4.
4.4. Abla ion s udy
Ou p oposed c i e ion (7) con ains h ee main e ms: he
g aph cons uc ion and i s egula iza ion, he smoo hness e m
and he con olu ion e m. To illus a e he ele ance o he p o-
posed c i e ion and i s e ms, we gene a e ou di e en mod-
els wi h di e en combina ions. These ou di e en a ian s o
MVCGE a e: MVCGE-G, MVCGE-S, MVCGE-C and MVCGE-SC. (1)
No g aph egula iza ion e m in he global objec i e unc ion (7)
(i.e., αis se o ze o), and we call he ob ained me hod MVCGE-G,
which means ha only he smoo hness and con olu ion e ms in
MVCGE a e used, (2) No smoo hness cons ain (λ1is se o ze o),
and we call he ob ained me hod MVCGE-S, (3) No con olu ion
e m (λ2is se o ze o), and we call he ob ained me hod MVCGE-
C, and (4) No smoo hness and no con olu ion e ms (λ1and λ2
a e se o ze o). This me hod is called MVCGE-SC because i is
educed o a consis en g aph cons uc ion ollowed by a spec al
clus e ing s ep. The esul s ob ained wi h MVCGE-G, MVCGE-S,
MVCGE-C, MVCGE-SC and MVCGE a e summa ized in Table 5. We
used h ee da ase s: ORL, MSRC 1 and Te a. F om he esul s in
Table 5, we can see ha he egula iza ion o he g aph is indeed
c ucial, since he las wo e ms depend on his g aph. Fo he
ORL and Te a da ase s, i can be seen om he able ha he
8
S. El Hajja , F. Do naika, F. Abdallah e al. Knowledge-Based Sys ems 241 (2022) 108250
Table 3
Clus e ing pe o mance on he BBCSpo , MSRC 1, Ex ended-Yale, MNIST and MNIST-1000 da ase s.
Da ase Me hod ACC NMI Pu i y ARI
MVCSK [19] 0.90 (±0.07) 0.82 (±0.02) 0.90 (±0.02) 0.85 (±0.07)
BBCSpo NESE [18] 0.72 (±0.00) 0.69 (±0.00) 0.75 (±0.00) 0.60 (±0.00)
S-MVSC [47] 0.58 (±0.07) 0.67 (±0.01) 0.73 (±0.02) 0.83 (±0.04)
CI-GMVC [48] 0.61 (±0.00) 0.46 (±0.00) 0.63 (±0.00) 0.36 (±0.00)
MCLES [49] 0.88 (±0.00) 0.80 (±0.00) 0.88 (±0.00) 0.83 (±0.00)
CNESE [9] 0.72 (±0.00) 0.68 (±0.00) 0.76 (±0.00) 0.60 (±0.00)
MVCGE 0.98 (±0.00)0.94 (±0.00)0.98 (±0.00)0.95 (±0.00)
MVCSK [19] 0.70 (±0.02) 0.59 (±0.03) 0.70 (±0.02) 0.50 (±0.04)
MSRC 1 NESE [18] 0.77 (±0.00) 0.72 (±0.00) 0.80 (±0.00) 0.64 (±0.00)
S-MVSC [47] 0.60 (±0.00) 0.69 (±0.02) 0.74 (±0.02) 0.79 (±0.01)
CI-GMVC [48] 0.74 (±0.00) 0.72 (±0.00) 0.77 (±0.00) 0.59 (±0.00)
MCLES [49] 0.90 (±0.01) 0.83 (±0.02) 0.90 (±0.01) 0.77 (±0.00)
CNESE [9] 0.86 (±0.00) 0.76 (±0.00) 0.86 (±0.00) 0.72 (±0.00)
MVCGE 0.93 (±0.00)0.87 (±0.00)0.93 (±0.00)0.85 (±0.00)
MVCSK [19] 0.33 (±0.00) 0.42 (±0.00) 0.34 (±0.00) 0.18 (±0.00)
Ex ended- NESE [18] 0.43 (±0.00) 0.58 (±0.00) 0.47 (±0.00) 0.25 (±0.00)
Yale S-MVSC [47] 0.48 (±0.03) 0.61 (±0.01) 0.60 (±0.01) 0.36 (±0.05)
CI-GMVC [48] 0.32 (±0.00) 0.34 (±0.00) 0.35 (±0.00) 0.02 (±0.00)
MCLES [49] 0.48 (±0.03) 0.48 (±0.00) 0.48 (±0.01) 0.10 (±0.05)
CNESE [9] 0.60 (±0.00) 0.75 (±0.00) 0.60 (±0.00) 0.51 (±0.00)
MVCGE 0.88 (±0.00)0.86 (±0.00)0.88 (±0.00)0.77 (±0.00)
MVCSK [19] 0.49 (±0.00) 0.41 ( ±0.00) 0.50 (±0.00) 0.29 (±0.00)
MNIST NESE [18]0.81 (±0.00)0.83 (±0.00) 0.85 (±0.00) 0.76 (±0.00)
S-MVSC [47] 0.77 (±0.01) 0.81 ( ±0.01) 0.81 (±0.02) 0.76 (±0.07)
CI-GMVC [48] 0.66 (±0.00) 0.71 ( ±0.00) 0.71 (±0.00) 0.51 (±0.00)
MCLES [49] 0.80 (±0.00) 0.83 (±0.00) 0.85 (±0.00) 0.77 (±0.00)
CNESE [9]0.81 (±0.00) 0.83 (±0.00) 0.86 (±0.00)0.78 (±0.00)
MVCGE 0.81 (±0.00)0.83 (±0.00) 0.85 (±0.00) 0.77 (±0.00)
MVCSK [19] 0.70 (±0.00) 0.61 ( ±0.00) 0.70 (±0.00) 0.52 (±0.00)
MNIST-1000 NESE [18] 0.78 (±0.00) 0.79 (±0.00) 0.83 (±0.00) 0.71 (±0.00)
S-MVSC [47] 0.66 (±0.02) 0.76 ( ±0.01) 0.76 (±0.00) 0.77 (±0.05)
CI-GMVC [48] 0.65 (±0.00) 0.71 (±0.00) 0.73 (±0.00) 0.50 (±0.00)
MCLES [49] 0.73 (±0.02) 0.72 (±0.01) 0.77 (±0.02) 0.58 (±0.04)
CNESE [9] 0.77 (±0.00) 0.77 (±0.00) 0.81 (±0.00) 0.68 (±0.00)
MVCGE 0.86 (±0.00)0.83 (±0.00)0.86 (±0.00)0.78 (±0.00)
Table 4
Clus e ing pe o mance on he h ee syn he ic da ase s.
Da ase Me hod ACC NMI Pu i y ARI
NESE [18] 0.64 0.75 0.75 0.63
Te a MVCSK [19] 0.97 0.93 0.97 0.92
S-MVSC [47] 0.70 0.50 0.44 0.70
CI-GMVC [48] 0.63 0.52 0.67 0.43
MCLES [49] 0.85 0.88 0.89 0.80
CNESE [9] 0.66 0.62 0.75 0.54
MVCGE 1.00 1.00 1.00 1.00
NESE [18] 0.81 0.79 0.85 0.73
Hep a MVCSK [19] 0.89 0.85 0.89 0.80
S-MVSC [47] 0.66 0.63 0.47 0.70
CI-GMVC [48] 0.77 0.76 0.81 0.68
MCLES [49] 0.87 0.82 0.84 0.80
CNESE [9] 0.78 0.70 0.79 0.63
MVCGE 0.92 0.85 0.92 0.83
NESE [18] 0.93 0.69 0.93 0.73
Chainlink MVCSK [19] 0.63 0.05 0.63 0.07
S-MVSC [47] 0.67 0.14 0.78 0.12
CI-GMVC [48] 0.55 0.01 0.55 0.01
MCLES [49] 0.90 0.72 0.86 0.76
CNESE [9] 0.95 0.70 0.95 0.78
MVCGE 0.96 0.78 0.96 0.85
smoo hness e m has a la ge impac on he clus e ing esul s.
Howe e , o he MSRC 1 da ase , he con olu ion e m is mo e
impo an han he smoo hness e m. This is no mal and is due
o he di e en ypes o da ase s used in his wo k. The esul s
ob ained wi h MVCGE-SC show he impo ance o he las wo
e ms in he objec i e unc ion o all da ase s. All hese esul s
indica e ha he inclusion o all e ms in he objec i e unc ion
Table 5
Abla ion s udy wi h di e en models. The bes pe o mance o each indica o
is in bold.
Da ase Va ian ACC NMI Pu i y ARI
MVCGE-G 0.46 0.66 0.48 0.27
ORL MVCGE-S 0.75 0.88 0.76 0.72
MVCGE-C 0.86 0.94 0.88 0.81
MVCGE-SC 0.69 0.86 0.75 0.57
MVCGE 0.93 0.97 0.95 0.92
MVCGE-G 0.68 0.57 0.68 0.45
MSRC 1 MVCGE-S 0.72 0.63 0.74 0.54
MVCGE-C 0.70 0.60 0.70 0.50
MVCGE-SC 0.59 0.54 0.61 0.37
MVCGE 0.93 0.87 0.93 0.85
MVCGE-G 0.70 0.59 0.72 0.55
Te a MVCGE-S 0.91 0.79 0.91 0.77
MVCGE-C 0.97 0.93 0.97 0.92
MVCGE-SC 0.56 0.35 0.57 0.33
MVCGE 1.00 1.00 1.00 1.00
con ibu ed o he good clus e ing pe o mance o ou p oposed
me hod.
4.5. Analysis o esul s and me hod compa ison
Acco ding o Table 2, he pe o mance o all mul i- iew clus-
e ing me hods is be e han ha o SC-Bes , which co esponds
o he spec al clus e ing me hod applied o he bes single iew.
In ac , he p esence o mul iple iews b ings addi ional in o ma-
ion o he clus e ing me hod so ha i can p ocess he da ase s
be e . The p oposed me hod gi es he bes pe o mance ol-
lowed by NESE, MVCSK, S-MVSC, CI-GMVC, MCLES and CNESE
9
