Comp es Rendus
Mécanique
Ruben Ibanez, Pie e Gilo mini, Elias Cue o and F ancisco Chines a
Nume ical expe imen s on unsupe ised mani old lea ning applied o
mechanical modeling o ma e ials and s uc u es
Volume 348, issue 10-11 (2020), p. 937-958.
<h ps://doi.o g/10.5802/c meca.53>
Pa o he Thema ic Issue: Con ibu ions in mechanics o ma e ials
Gues edi o s: Julie Diani, Oli ie Cas elnau and F ancisco Chines a
© Académie des sciences, Pa is and he au ho s, 2020.
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Comp es Rendus
Mécanique
2020, 348, n10-11, p. 937-958
h ps://doi.o g/10.5802/c meca.53
Con ibu ions in mechanics o ma e ials
Nume ical expe imen s on unsupe ised
mani old lea ning applied o mechanical
modeling o ma e ials and s uc u es
Ruben Ibaneza, Pie e Gilo minia, Elias Cue oband F ancisco Chines a∗,a
aPIMM lab, A s e Me ie s Ins i u e o Technology, 151 Boule a d de Hôpi al, 75013
Pa is, F ance
bA agon Ins i u e o Enginee ing Resea ch, Uni e sidad de Za agoza, Ma ia de Luna
s/n, 50018 Za agoza, Spain
E-mails: [email p o ec ed] (R. Ibanez), pie [email p o ec ed]
(P. Gilo mini), ecue o@uniza .es (E. Cue o), ancisco[email p o ec ed]
(F. Chines a)
Abs ac . The p esen wo k aims a analyzing issues ela ed o he da a mani old dimensionali y. The in e es
o he s udy is wo old: (i) i s , when oo many measu able a iables a e conside ed, mani old lea ning is
expec ed o ex ac useless a iables; (ii) second, and mo e impo an , he same echnique, mani old lea ning,
could be u ilized o iden i ying he necessi y o employing la en ex a a iables able o eco e single-
alued ou pu s. Bo h aspec s a e discussed in he modeling o ma e ials and s uc u al sys ems by using
unsupe ised mani old lea ning s a egies.
Keywo ds. Nonsupe ised mani old lea ning, S a e a iables, Dimensionali y educ ion, k-PCA, S uc u al
analysis, Ma e ial cons i u i e equa ions.
Manusc ip ecei ed 21s June 2020, e ised 12 h July 2020, accep ed 7 h Oc obe 2020.
1. In oduc ion
Recen ly, da a-d i en desc ip ion o ma e ials has been gaining popula i y. Many complex ma-
e ial beha io s esis ing adi ional modeling p ocedu es, o ha a e oo complex om a mi-
c os uc u al iewpoin , a e app oached by using da a-based desc ip ions. Di e en app oaches
a e being conside ed. Among hem include hose based exclusi ely on measu ed da a, o he s
ha ex ac he mani olds ela ed o da a, and o he s ha a emp o en o ce he modynamic
and he momechanical consis ency. The in e es ed eade can e e o [1–7] and he nume ous
e e ences he ein. This wo k ocuses on echniques based on he use o mani olds and hei as-
socia ed mani old lea ning p ocedu es o ex ac ing hem om he a ailable da a.
∗Co esponding au ho .
ISSN (elec onic) : 1873-7234 h ps://comp es- endus.academie-sciences. /mecanique/
938 Ruben Ibanez e al.
Figu e 1. Mul idimensional da a on one- (le ), wo- (cen e ), and h ee-dimensional
( igh ) mani olds embedded in RD.
In gene al, da a in ol e many dimensions. Conside i s a sequence o h ee-dimensional
(3D) ields de ined in a domain Ω⊂R3pa i ioned in o D oxels. Each o hese ields con ains
many da a, one da um a each oxel. Each ield can be ep esen ed as a poin in a ec o space
o dimension D( he numbe o oxels), whe e we can p esume each o he Dcoo dina e axes
as epo ing he alue ha he ield o in e es akes in he associa ed oxel. Thus, each ield
becomes a poin in ha high-dimensional space o dimension D,RD. I impo an co ela ions
exis among he di e en ields, hese poin s a e expec ed o be dis ibu ed on a low-dimensional
subspace embedded in he D-dimensional space. Techniques aiming a ex ac ing hese educed
subspaces, he so-called slow mani olds, ske ched in Figu e 1, a e key ools o manipula ing da a
and ex ac ing hei hidden in o ma ion.
Thus, da a de ine in gene al slow mani olds embedded in e y la ge ec o spaces due o he
signi ican hidden co ela ions among hem. The numbe o unco ela ed explica i e dimensions
usually becomes much smalle han he a p io i assumed dimension o he space o accommo-
da ing he da a. The ex ac ion o hese slow mani olds can be success ully accomplished by using
linea and nonlinea dimensionali y educ ion echniques such as p incipal componen analysis
(PCA) in he linea case and i s nonlinea coun e pa s (`PCA, ke nel-based PCA [k-PCA], LLE,
SNE, e c.) [8–12].
These echniques can be applied o se e al physical sys ems. Mo eo e , when he slow mani-
old is de e mined, he solu ion a any poin on i can be compu ed e y accu a ely om a sim-
ple in e pola ion o neighbo ing da a on he mani old [13], enabling almos eal- ime p edic ions
and he associa ed eal- ime decision-making.
Howe e , ex ac ing knowledge om da a associa ed wi h an exis ing bu hidden model
equi es he ollowing:
•iden i ying he mani old in insic dimension,
•disco e ing hidden pa ame e s,
•disca ding useless pa ame e s, and
•disco e ing he models o igina ing he da a.
These ques ions a e add essed, illus a ed, and discussed in he p esen wo k in a pu ely
me hodological manne , aiming a illus a ing he key concep s ha could open nume ous u u e
possibili ies in he ield o mechanics o ma e ials, p ocesses, s uc u es, and sys ems.
In ou p e ious wo ks, we add essed p oblems in ol ing housands o dimensions [13–16],
p o ing ha despi e he appa en ichness in many cases, he embedding ega ds a ela i ely
low-dimensional space. Howe e , in mos o he p oblems ha we ha e ea ed un il now,
hei complexi y p e en ed ine analyses o hei solu ions. The p esen pape , which is pu ely
me hodological, conside s simple p oblems de ined in low dimensions, wi h known solu ions
C. R. Mécanique,2020, 348, n10-11, 937-958
Ruben Ibanez e al. 939
being easily isualizable, o acili a ing an analysis and discussion. O cou se, and as p o ed in
he wo ks jus e e ed o, all he me hodologies apply o mul idimensional se ings.
2. Unsupe ised mani old lea ning
Le us conside a ec o y∈RDcon aining expe imen al o syn he ic da a om measu emen s
o nume ical simula ion. These esul s a e o en e e ed o as snapsho s. I hey a e ob ained
by nume ical simula ion, hey consis o nodal alues o he essen ial a iable. The e o e, hese
a iables will be somehow co ela ed and, no ably, he e will be a linea ans o ma ion W
de ining he ec o ξ∈Rd, wi h d<D, which con ains he s ill unknown la en a iables such
ha
y=Wξ. (1)
The D×d ans o ma ion ma ix W, which sa is ies he o hogonali y condi ion WTW=Id,
is he main ing edien o he PCA and can be compu ed as de ailed in Appendix A om he
co a iance ma ix associa ed wi h a numbe (M) o snapsho s y1,...,yM, which cons i u e he
columns o ma ix Y.
While PCA wo ks wi h he co a iance ma ix (i.e., YYT), mul idimensional scaling (MDS) wo ks
wi h he G am ma ix con aining scala p oduc s (i.e., S=YTY) as desc ibed in Appendix A.
On he o he hand, he k-PCA is based on he ac ha da a no linea ly sepa able in D
dimensions could be linea ly sepa a ed i p e iously p ojec ed o a space in Q>Ddimensions.
Howe e , he ue ad an age a ises om he ac ha i is no necessa y o w i e down he
analy ical exp ession o ha mapping as desc ibed in Appendix A.
3. An illus a i e s uc u al mechanics case s udy
Conside i s a hypo he ical mechanical sys em consis ing o a p isma ic beam whose h ee
dimensions, heigh , wid h, and leng h, a e deno ed, espec i ely, by h,b, and L, all o hem being
measu able quan i ies. In wha ollows, we conside a pa icula ou pu P ha cons i u es also a
measu able quan i y (buckling c i ical load, e c.) assumed ela ed o hose pa ame e s om an
exis ing bu ac ually hidden model e en i in wha ollows we will conside hypo he ical, and mos
o he ime, unphysical models.
Thus, we conside a se o da a composed o Mmeasu es yi={hi,bi,Li,Pi}, i=1,...,M, wi h
hi,bi, and Libeing andomly chosen om a uni o m p obabili y dis ibu ion in hei espec i e
in e als o exis ence, Ih,Ib, and IL, espec i ely, de ined om he ollowing:



Ih=[hmin,hmax]
Ib=[bmin,bmax]
IL=[Lmin,Lmax].
(2)
Wi hou any o he p e-exis ing knowledge, one expec s he ou pu depending on he h ee ge-
ome ical pa ame e s (i.e., P=P(h,b,L)). In wha ollows, we conside h ee di e en scena ios.
3.1. Ou pu depending on a single pa ame e
In his sec ion, we assume a qui e simple model ha ela es he ou pu o a single pa ame e ,
P=αb,α∈R+, (3)
whe e α=1000 in he nume ical es s ca ied ou . We pe o m M=1000 measu es, whe e
Ih∈[0.1,0.2], Ib∈[0.15,0.2], and IL∈[1,1.5].
C. R. Mécanique,2020, 348, n10-11, 937-958
940 Ruben Ibanez e al.
Figu e 2. Eigen alues λ∈[λ110−6,λ1].
The measu es cons i u e a se o M=1000 poin s in R4on which he k-PCA is applied by using
he Gaussian ke nel
κ(yi,yj)=exp−
kyi−yjk2
2β2, (4)
whe e β=10.
Figu e 2 depic s he highes eigen alues among M esul ing om he k-PCA, hose lying
be ween he highes alue λ1and 10−6λ1.
The slow mani old associa ed wi h ξis ep esen ed by selec ing he i s h ee educed coo -
dina es (ξ1,ξ2,ξ3) as shown in Figu e 3, whe e i s one-dimensional (1D) in insic dimension is
no ed. This esul was expec ed om he conside ed model exp essed by (3). The poin s on he
slow mani old a e colo ed depending on he alues o h,b,L, and P, e idencing ha bcons i u es
he la en a iable and ha he ou pu Pscales ( isually) linea ly wi h i .
The p ocess o colo ing he da a poin s in he embedded mani old dese es some addi ional
commen s due o he ac ha his is used in all he analyses epo ed in he p esen pape . Man-
i old lea ning echniques look o a low-dimensional mani old de ined by da a poin s. As soon as
he slow mani old is ex ac ed, he di e en da a poin s can be mapped on i . This isualiza ion
is only possible when he numbe o dimensions allows a simple g aphical ep esen a ion (as is
he case o he p oblems add essed in he p esen pape ). Then, hese poin s can be colo ed de-
pending on he alue o he di e en ini ial coo dina es, and one expec s ha i he e is a co e-
la ion (di ec o in e se and linea o nonlinea ) be ween he ini ial and educed coo dina es, he
colo s mus exhibi a ce ain g ading.
E en i his analysis seems qui e dependen on he low dimensionali y o he embedding, in
highe dimensional embeddings, he analysis can be pe o med by using local s a is ics. Thus,
by conside ing a da a poin in he slow mani old and i s closes neighbo s, a local s a is ical
analysis can be easily pe o med wi h he s anda d de ia ion indica ing he dispe sion o he
da a (equi alen o he local dispe sion o colo s).
One could be sligh ly su p ised by he nonlinea i y ha he mani old exhibi s despi e he
linea i y o model (3). This nonlinea i y is an a i ac o he nonlinea ke nel (4) used. As he model
is linea , one could expec he abili y o he PCA o add ess he p oblem a hand. Fo his pu pose,
i su ices ans o ming he ke nel in o i s linea coun e pa , gi ing ise o he PCA
κ(yi,yj)=yi·yj. (5)
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Ruben Ibanez e al. 941
Figu e 3. Slow mani old in he 3D space de ined by he i s educed coo dina es (ξ1,ξ2,ξ3).
Each poin is colo ed acco ding o he alue o he coo dina e h( op le ), b( op igh ), L
(bo om le ), o P(bo om igh ).
Figu e 4. Slow mani old ξwhen conside ing he PCA linea dimensionali y educ ion. Each
poin is colo ed acco ding o he alue o he a iable b.
In his case, as expec ed, a single nonze o eigen alue esul s, and consequen ly he dimension
o he educed space becomes one (i.e., ξ=ξ). The associa ed mani old is depic ed in Figu e 4.
Now, we conside a sligh ly di e en model, again depending on a single a iable bu in a
nonlinea manne , acco ding o
P=αh2, (6)
C. R. Mécanique,2020, 348, n10-11, 937-958
942 Ruben Ibanez e al.
Figu e 5. Slow mani old in he 3D space de ined by he i s h ee educed coo dina es
(ξ1,ξ2,ξ3). Each poin is colo ed acco ding o he alue o he coo dina e h(le ) o P( igh ).
Figu e 6. Slow mani old in he 3D space de ined by he i s h ee educed coo dina es
(ξ1,ξ2,ξ3). Each poin is colo ed acco ding o he alue o he coo dina e L(le ) o P( igh ).
whe e again α=1000. Figu e 5 depic s he 1D slow mani old, whe e poin s a e colo ed acco ding
o he alues o he a iables hand P. He e, e en i he di ec ela ion can be no ed, i s nonlinea -
i y is much less e iden o isualize.
Finally, we conside he model
P=α
L2, (7)
whe e α=1000. Figu e 6 depic s he 1D slow mani old, whe e poin s a e colo ed acco ding o he
alues o he a iables Land P o emphasize he in e se ela ion be ween hem.
3.2. Ou pu depending on wo pa ame e s
In his sec ion, we conside a model in ol ing wo o he h ee a iables, in pa icula ,
P=αh3
L2, (8)
whe e α=1000.
Figu e 7 depic s he wo-dimensional (2D) slow mani old, whe e poin s a e colo ed acco ding
o he alues o a iables h,b,L, and P. He e, he di ec and in e se e ec s o hand Lwi h espec
o Pcan be no ed as well as he ac ha pa ame e bseems, and in ac is, useless.
C. R. Mécanique,2020, 348, n10-11, 937-958
Ruben Ibanez e al. 943
Figu e 7. Slow mani old in he 3D space de ined by he i s h ee educed coo dina es
(ξ1,ξ2,ξ3). Each poin is colo ed acco ding o he alue o he coo dina e h( op le ), b( op
igh ), L(bo om le ), o P(bo om igh ).
3.3. Iden i ying hidden a iables
The p e ious case s udies e ealed he abili y o ex ac he in insic dimensionali y o he slow
mani old as well as he possibili y o iden i y useless pa ame e s. The p esen case add esses a
e y di e en si ua ion in which he model in ol es he h ee a iables in he discussed case, h,
b, and L. Howe e , only wo o hem we e measu ed, namely hand Lwi h he ou pu P, wi h b
emaining inaccessible.
Thus, we ha e
P=αbh3
L2, (9)
whe e α=1000. The e o e, he M=1000 collec ed da a yi,i=1,...,M, eads as yi={hi,Li,Pi}∈R3.
Figu e 8 depic s he educed poin s (ξ1,ξ2,ξ3), which as can be seen a e dis ibu ed in a
domain ω⊂R3. Howe e , no dimensionali y educ ion is no ed, and he embedding emains
3D. A di ec consequence is ha many alues o he ou pu Pexis o he same alues on he
measu ed inpu s hand L, ela ed o he di e en alues o b, which a ec he ou pu P. Howe e ,
as bis no measu ed, i s alue is no conside ed in he da a poin s.
Such a mul i alued ou pu does no ep esen any concep ual di icul y. I indica es ha e en
i bo h a iables pa icipa e in he ou pu , he e may be o he s ha we e no conside ed o hose
conside ed may be useless o explaining he ou pu .
C. R. Mécanique,2020, 348, n10-11, 937-958
944 Ruben Ibanez e al.
Figu e 8. Reduced ep esen a ion ξi∈R3o he da a yi∈R3, whe e he educed poin s a e
colo ed acco ding o he alue o he ou pu P.
Figu e 9. Reduced poin s colo ed acco ding o he alues o he coo dina es h(le ) and L
( igh ).
To conclude abou he pe inence o hese a iables wi h espec o he conside ed ou pu ,
we conside colo ing he educed da a depending on he hand L alues ( e e o Figu e 9). We
compa e hem wi h he one whe e he colo scales wi h he ou pu P epo ed in Figu e 8.
Thus, one could conclude ha bo h a iables hand La e ele an o explaining he ou pu P.
I we assume ha he ou pu Pshould be uni ocally explained om a small numbe o a iables,
clea ly only wo a iables (he e hand L) a e no su icien . One ex a dimension su ices o
eco e ing a single- alued ou pu , ha is, conside ing he educed poin s in ou dimensions R4.
As isualizing hings in ou dimensions is qui e a di icul ask, o he sake o cla i y in he
exposi ion, in wha ollows, we p opose add essing a simple model in ol ing lowe dimensional
spaces.
We conside he simple model
P=αbh3, (10)
whe e α=1000. Howe e , he M=1000 collec ed da a yi,i=1,...,M, only deal wi h hand P, (i.e.,
yi={hi,Pi}∈R2).
Figu e 10 depic s he da ase yi={hi,Pi}, whe e i can be no ed ha many alues o he ou pu
Pa e ound o he same alue o he a iable h.
Fo uni ocally exp essing he ou pu P, we conside again he k-PCA. We compu e he educed
da ase in a 3D space by conside ing he i s h ee coo dina es (ξ1,ξ2,ξ3), while colo ing hese
C. R. Mécanique,2020, 348, n10-11, 937-958
Ruben Ibanez e al. 951
Figu e 22. Slow mani old ela ed o linea elas ic beha io and colo ed acco ding o he
angen modulus (i.e., ∆σi/∆εi).
Figu e 23. Slow mani old ela ed o nonlinea elas ic beha io and colo ed acco ding o he
angen modulus (i.e., ∆σi/∆εi).
ins an aneous angen modulus when loading and unloading. As Figu e 26 e eals, he
mechanical mani old is now iche wi h poin s on he elas ic domain bounda y sepa a ed
om he elas ic mani old. In he solu ion depic ed in Figu e 26, damage is no ac i a ed.
In he p esence o damage, he educ ion in elas ic angen modulus scales wi h he plas-
ic s ain. The slow mani old is colo ed acco ding o he angen modulus as shown in
Figu e 27.
•The las scena io adds an ex a ichness o he cons i u i e beha io . A he mechani-
cal s a es wi hin he elas ic domain, he elas ic angen modulus is a ec ed by a la en
C. R. Mécanique,2020, 348, n10-11, 937-958

952 Ruben Ibanez e al.
Figu e 24. S ain–s ess poin s associa ed wi h s a es wi hin he elas ic domain wi h
nonze o plas ic s ains (le ) and he associa ed slow mani old colo ed acco ding o he an-
gen modulus ( igh ).
Figu e 25. Slow mani old ela ed o damageable elas ic–plas ic beha io ope a ing wi hin
he elas ic domain colo ed acco ding o he angen modulus.
a iable ha is he p oduc o he plas ic s ain wi h ano he ex a a iable. The la e ,
which could ep esen s ain- a e sensi i i y, is assumed o ake a bi a y alues he e due
o he ac ha we a e mo e in e es ed in me hodological aspec s han in physical con-
side a ions. The conside ed mechanical s a es a e depic ed in Figu e 28. When apply-
ing he k-PCA o he se o mechanical s a es yi={εi,σi,∆σi/∆εi}, he esul ing mani-
old emains 3D. As expec ed, no dimensionali y educ ion is accomplished as Figu e 29
e eals, whe e many alues o he elas ic angen modulus can be ound o he same
alues o he s ess and s ain. To in es iga e he na u e o his beha io , we depic in Fig-
u e 30 he elas ic angen modulus e sus he plas ic s ain, whe e as expec ed i can be
no ed ha he o me does no depend exclusi ely on he la e . By applying he k-PCA
o he da a shown in Figu e 30, which a e nonsepa able in wo dimensions, one expec s
C. R. Mécanique,2020, 348, n10-11, 937-958
Ruben Ibanez e al. 953
Figu e 26. Slow mani old ela ed o elas ic–plas ic beha io ope a ing wi hin he elas ic
domain and on he elas ic domain bounda y colo ed acco ding o he angen modulus.
Figu e 27. Slow mani old ela ed o elas ic–plas ic beha io ope a ing wi hin he elas ic
domain and on he elas ic domain bounda y, colo ed acco ding o he angen modulus,
when damage is ac i a ed.
o sepa a e hem by embedding in a 3D space as p e iously discussed and as Figu e 31
p o es. Finally, Figu e 32 p esen s he slow mani old om Figu e 31 bu now colo ed wi h
espec o he plas ic s ain o he ex a la en a iable.
C. R. Mécanique,2020, 348, n10-11, 937-958
954 Ruben Ibanez e al.
Figu e 28. Mechanical s a es wi hin he elas ic domain, colo ed acco ding o he angen
modulus, he las depending on he p oduc o wo la en a iables, he plas ic s ain, and
ano he a bi a ily chosen a iable.
Figu e 29. Slow mani old ela ed o elas ic–plas ic beha io ope a ing wi hin he elas ic
domain, colo ed acco ding o he angen modulus, wi h he las scaling wi h he p oduc
o he plas ic s ain and an ex a la en a iable.
Figu e 30. Elas ic angen modulus e sus he plas ic s ain.
C. R. Mécanique,2020, 348, n10-11, 937-958
Ruben Ibanez e al. 955
Figu e 31. Slow mani old ela ed o he da a consis ing o he elas ic angen modulus and
he plas ic s ain.
Figu e 32. Slow mani old o he elas ic angen modulus colo ed acco ding o he plas ic
s ain (le ) and he ex a la en a iable ( igh ).
6. Conclusion
The p esen wo k in oduced, es ed, and discussed issues ela ed o mani old dimensionali y
wi h wo majo pu poses: (i) i s , when oo many measu able a iables a e employed, mani old
lea ning is able o disca d he useless a iables; (ii) second and mo e impo an , he same
echnique can be employed o disco e ing he necessi y o employing and hen measu ing an
ex a la en a iable ha is able o eco e and ensu e single- alued ou pu s. Bo h issues we e
analyzed and discussed in wo case s udies, one wi h espec o s uc u al mechanics and he
o he wi h espec o pa h-dependen ma e ial cons i u i e beha io s.
Indeed, he physical in e p e a ion o he disco e ed la en a iable could imply he in oduc-
ion o o he measu able a iables. This opic should be deeply analyzed.
The main in e es in disco e ing hese mani olds is ha o a new accessible mechanical s a e,
he ou pu can be in e ed by a simple in e pola ion om i s neighbo s on he mani old. The
main aim o he p esen wo k is he cons uc ion and analysis o hese mani olds. Fu u e wo ks,
cu en ly in p og ess, will ocus on hei use in pe o ming da a-d i en simula ions.
Con lic s o in e es
The au ho s decla e no compe ing inancial in e es s.
C. R. Mécanique,2020, 348, n10-11, 937-958
956 Ruben Ibanez e al.
Dedica ion
This manusc ip was w i en wi h he help o he con ibu ions o all au ho s. All au ho s ha e
gi en app o al o he inal e sion o he manusc ip .
Acknowledgmen s
The i s , hi d, and ou h au ho s a e suppo ed by hei espec i e ESI G oup esea ch chai s;
hei suppo is g a e ully acknowledged. The i s au ho is suppo ed by CREATE-ID ESI-ENSAM
esea ch chai . The hi d au ho is suppo ed by he ESI G oup Chai a he Uni e si y o Za agoza.
The ou h au ho is suppo ed by CREATE-ID ESI-ENSAM esea ch chai .
Appendix A. F om p incipal componen analysis o i s ke nel-based coun e pa
A.1. P incipal componen analysis
Le us conside a ec o y∈RDcon aining expe imen al esul s o syn he ic da a om a nume i-
cal simula ion. These esul s a e o en e e ed o as snapsho s. I hey a e ob ained by nume ical
simula ion, hey consis o nodal alues o he essen ial a iable. The e o e, hese a iables will
be somehow co ela ed and, no ably, he e will be a linea ans o ma ion Wde ining he ec o
ξ∈Rd, wi h d<D, which con ains he s ill unknown la en a iables, such ha
y=Wξ. (11)
The D×d ans o ma ion ma ix W, which sa is ies he o hogonali y condi ion WTW=Id, is
he main ing edien o he PCA [8].
Assume ha he e exis Mdi e en snapsho s y1,...,yM, which we s o e in he columns o a D×M
ma ix Y. The associa ed d×M educed ma ix Ξcon ains he associa ed ec o s ξi,i=1,...,M.
The PCA usually wo ks wi h cen e ed a iables. In o he wo ds,









M
X
i=1
yi=0,
M
X
i=1
ξi=0,
(12)
implying he necessi y o cen e ing da a be o e applying he PCA.
The PCA p oceeds by gua an eeing maximal p ese ed a iance and minimal co ela ion in
he la en a iable se ξ. The la en a iables in ξa e he e o e unco ela ed, and consequen ly
he co a iance ma ix o ξ,
Cξξ =E{ΞΞT}, (13)
should be diagonal.
To ex ac he dunco ela ed la en a iables, we p oceed om
Cyy =E{YYT}=E{WΞΞTWT}=WE{ΞΞT}WT=WCξξWT. (14)
P e- and pos -mul iplying by WTand W, espec i ely, and making use o he ac ha WTW=I,
gi e us
Cξξ =WTCy y W. (15)
The co a iance ma ix Cy y can hen be ac o ized by applying he singula alue decomposi-
ion,
Cyy =VΛVT, (16)
C. R. Mécanique,2020, 348, n10-11, 937-958

Ruben Ibanez e al. 957
whe e Vcon ains he o hono mal eigen ec o s; Λis a diagonal ma ix con aining he eigen al-
ues so ed in descending o de .
Subs i u ing (16) in o (15), we a i e a
Cξξ =WTVΛVTW. (17)
This equali y holds when he dcolumns o Wa e aken o be collinea wi h dcolumns o V.
We hen p ese e he eigen ec o s associa ed wi h he dnonze o eigen alues,
W=VID×d, (18)
which gi es
Cξξ =Id×DΛID×d. (19)
We he e o e conclude ha he eigen alues in Λ ep esen he a iance o he la en a iables
(diagonal en ies o Cξξ).
A.2. Mul idimensional scaling
The PCA wo ks wi h he co a iance ma ix o he expe imen al esul s, YYT. Howe e , he MDS
wo ks wi h he G am ma ix con aining scala p oduc s (i.e., S=YTY) [8].
The MDS p ese es pai wise scala p oduc s:
S=YTY=ΞTWTWΞ=ΞTΞ. (20)
Compu ing he eigen alues o S, we a i e a
S=UΛUT=(UΛ1/2)(Λ1/2UT)=(Λ1/2UT)T(Λ1/2UT), (21)
which in u n gi es
Ξ=Id×MΛ1/2UT. (22)
A.3. Ke nel-based p incipal componen analysis
The k-PCA is based on he ac ha da a no linea ly sepa able in Ddimensions could be linea ly
sepa a ed i hey a e p e iously p ojec ed o a space in Q>Ddimensions. Howe e , he ue
ad an age a ises om he ac ha i is no necessa y o w i e down he analy ical exp ession o
ha mapping.
The symme ic ma ix Φ=ZTZ, wi h Zcon aining he snapsho s zi∈RQ,i=1,...,M, associa ed
wi h yi∈RD, has o be decomposed in o eigen alues and eigen ec o s. The p ocedu e o
cen e ing da a ziis ca ied ou in an implici way.
The eigen ec o decomposi ion eads as
Φ=UΛUT, (23)
gi ing ise o
Ξ=Id×MΛ1/2UT. (24)
The di icul ies o ope a ing in a high-dimensional space o dimension, in gene al, QÀD,
and he mapping una ailabili y a e ci cum en ed by in oducing he ke nel unc ional κ(also
known as he ke nel ick). This allows compu ing scala p oduc s in RQwhile ope a ing in RDby
applying he Me ce heo em. This heo em es ablishes ha i κ(u, ) (whe e u∈RDand ∈RD)
is con inuous, symme ic, and posi i e de ini e, hen i de ines an inne p oduc in he mapped
space RQ. Many di e en ke nels exis ; some o hem a e epo ed in [8].
C. R. Mécanique,2020, 348, n10-11, 937-958
958 Ruben Ibanez e al.
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