T ainable and explainable simplicial map neu al ne wo ks
Edua do Paluzo-Hidalgo a,∗, Rocio Gonzalez-Diazb, Miguel A. Gu ié ez-Na anjo c
aDepa men o Quan i a i e Me hods, Uni e sidad Loyola Andalucía, Campus Se illa, Dos He manas, Se ille, Spain
bDepa amen o de Ma emá ica Aplicada I, Uni e sidad de Se illa, Se ille, Spain
cDepa amen o de Ciencias de la Compu ación e In eligencia A ificial, Uni e sidad de Se illa, Se ille, Spain
A R T I C L E I N F O A B S T R A C T
Keywo ds:
T aining neu al ne wo k
Simplicial maps
Explainable a ificial in elligence
Simplicial map neu al ne wo ks (SMNNs) a e opology-based neu al ne wo ks wi h in e es ing
p ope ies such as uni e sal app oxima ion abili y and obus ness o ad e sa ial examples unde
app op ia e condi ions. Howe e , SMNNs p esen some bo lenecks o hei possible applica ion
in high-dimensional da ase s. Fi s , SMNNs ha e p ecompu ed fixed weigh and no SMNN aining
p ocess has been defined so a , so hey lack gene aliza ion abili y. Second, SMNNs equi e he
cons uc ion o a con ex poly ope su ounding he inpu da ase . In his pape , we o e come hese
issues by p oposing an SMNN aining p ocedu e based on a suppo subse o he gi en da ase
and eplacing he cons uc ion o he con ex poly ope by a me hod based on p ojec ions o a
hype sphe e. In addi ion, he explainabili y capaci y o SMNNs and effec i e implemen a ion a e
also newly in oduced in his pape .
1. In oduc ion
In ecen yea s, A ificial In elligence (AI) me hods in gene al and Machine Lea ning me hods in pa icula ha e eached success
in eal-li e p oblems ha we e unexpec ed only a ew yea s ago. Many diffe en a eas ha e con ibu ed o his de elopmen . Among
hem, we can ci e he esea ch on new heo e ical algo i hms, he inc easing compu a ional powe o he la es gene a ion ha dwa e,
and he apid access o a huge amoun o da a. Such a combina ion o ac o s leads o he de elopmen o inc easingly complex
sel - egula ed AI me hods.
Many AI models cu en ly used a e based on backp opaga ion algo i hms, which ain and egula e hemsel es o achie e a
goal, such as classifica ion, ecommenda ion, o p edic ion. These sel - egula ing models achie e some kind o knowledge as hey
success ully e alua e es da a independen o he da a used o ain hem. None heless, such knowledge is usually exp essed in a
non-human- eadable way.
To fill he gap be ween he ecen de elopmen o AI models and hei social use, many esea che s ha e ocused on he de elop-
men o Explainable A ificial In elligence (XAI), which consis s o a se o echniques o p o ide clea , unde s andable, anspa en ,
in elligible, us wo hy, and in e p e able explana ions o he decisions, p edic ions, and easoning p ocesses made by he AI models,
a he han jus p esen ing hei ou pu , especially in domains whe e AI decisions can ha e significan consequences on human li e.
A global axonomy o in e p e able AI wi h he aim o uni ying e minology o achie e cla i y and efficiency in he defini ion o
* Co esponding au ho .
E-mail add esses: [email p o ec ed] (E. Paluzo-Hidalgo), [email p o ec ed] (R. Gonzalez-Diaz), [email p o ec ed] (M.A. Gu ié ez-Na anjo).
URLs: h ps://pe sonal.us.es/ ogodi (R. Gonzalez-Diaz), h p://www.cs.us.es/~na anjo/ (M.A. Gu ié ez-Na anjo).
h ps://doi.o g/10.1016/j.ins.2024.120474
Recei ed 26 Oc obe 2023; Recei ed in e ised o m 22 Janua y 2024; Accep ed 11 Ma ch 2024
A ailable online 18 Ma ch 2024
0020-0255/© 2024 The Au ho (s).
(h p://c ea i ecommons.o g/licenses/by/4.0/).
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egula ions o he de elopmen o e hical and eliable AI can be ound in [1]. Mo eo e , a nice in oduc ion and gene al ision can
be ound in [2]. Ano he cla i ying pape wi h defini ions, concep s, and applica ions o XAI in [3].
The so-called Simplicial Map Neu al Ne wo ks (SMNNs) we e in oduced in [4]as a cons uc i e app oach o he p oblem o
app oxima ing a con inuous unc ion on a compac se in a iangula ed space. Since he o iginal aim o he defini ion o SMNNs was
ocused on building a cons uc i e app oach, he compu a ion o hei weigh s was no based on an op imiza ion p ocess, as usual in
neu al ne wo ks, bu on a de e minis ic calculus. The a chi ec u e o SMNNs and he compu a ion o he se o weigh s a e based
on a combina o ial opology ool called simplicial maps. Mo eo e , SMNNs can be used o classifica ion and can be cons uc ed o
be obus o ad e sa ial examples [5]. Besides, hei a chi ec u e can be educed while main aining accu acy [6], being in a ian o
ans o ma ion i he ans o ma ion p ese es he ba ycen ic coo dina es (scale, o a ion, symme ies, e c.). As defined in [4,6,5],
SMNNs a e buil as a wo-hidden-laye eed- o wa d ne wo k whe e he se o weigh s is p ecompu ed based on he calcula ion o a
iangula ed con ex poly ope su ounding he inpu da a. As o he app oxima ions o con inuous unc ions wi h a bi a y p ecision
(see, o example, [7]), SMNNs ha e fixed weigh s, which means ha he weigh s depend only on he iangula ion made wi h he
poin s o he da ase as he suppo se and no aining p ocess is applied.
Summing up, some o he limi a ions o he SMNNs un il now a e ha hey a e cos ly o calcula e since he numbe o neu ons
is p opo ional o he numbe o simplices o he iangula ion suppo ed on he inpu da ase , and hey suffe om o e fi ing and
he e o e no gene alize well. These aspec s make SMNNs no used in p ac ice so a , al hough he idea o ela ing simplicial maps o
neu al ne wo ks is dis up i e and p o ides a new b idge ha can en ich bo h a eas.
In his pape , we p opose a me hod o make SMNNs efficien by educing hei size (in e ms o he numbe o neu ons ha
depends on he e ices o he iangula ion) and ha success ully makes SMNNs ainable and wi h gene aliza ion abili y. Besides,
we also p esen a s udy o he selec ion o he e ices om which we ob ain he iangula ion. Al hough SMNNs conside he
e ices o a simplex as pa o he necessa y in o ma ion o he classifica ion ask, he app oach p esen ed in his pape is a om
he classic Machine Lea ning ins ance-based me hods. Such me hods ely on a de e minis ic compu a ion based on dis ances, bu ,
in he app oach p esen ed in his pape , he compu a ion o he weigh s is he esul o an op imiza ion me hod in a p obabili y
dis ibu ion space. Finally, om an XAI poin o iew, we will see in his pape ha SMNNs a e explainable models since all decision
s eps o compu e he ou pu o SMNNs a e unde s andable and anspa en , and he e o e us wo hy.
The pape is o ganized as ollows. Fi s , some concep s o compu a ional opology and he defini ion o SMNNs a e ecalled in
Sec ion 2. Nex , in Sec ion 3we de elop se e al echnical de ails needed o he SMNN aining p ocess, which will be in oduced
in Sec ion 4. Sec ion 5is de o ed o he explainabili y o he model. Sec ion 6is de o ed o discussion and limi a ions. Finally, he
pape ends wi h some expe imen s and conclusions.
2. Backg ound
In his sec ion, we assume ha he eade is amilia wi h he basic concep s o compu a ional opology. Fo a comp ehensi e
p esen a ion, we e e o [8].
2.1. Simplicial complexes
Conside a fini e se o poin s 𝑉={𝑣1, … , 𝑣𝛽} ⊂ℝ𝑛whose elemen s will be called e ices. A subse
𝜎=⟨𝑣𝑖0,𝑣
𝑖1,…,𝑣
𝑖𝑑⟩
o 𝑉wi h 𝑑+1 e ices (in gene al posi ion) is called a 𝑑-simplex. The con ex hull o he e ices o 𝜎will be deno ed by |𝜎|and
co esponds o he se :
{𝑥∈ℝ𝑛∶𝑥=∑
𝑗∈0,𝑑
𝑏𝑗(𝑥)𝑣𝑖𝑗}
whe e 𝑎, 𝑏 ={𝑎, 𝑎 +1, … , 𝑏} o 𝑎 <𝑏 ∈ℤ, and
𝑏(𝑥)=(𝑏0(𝑥),𝑏
1(𝑥),…,𝑏
𝑑(𝑥))
a e called he ba ycen ic coo dina es o 𝑥wi h espec o 𝜎, and sa is y ha :
∑
𝑗∈0,𝑑
𝑏𝑗(𝑥)=1 and 𝑏𝑗(𝑥)≥0∀𝑗∈0,𝑑.
The ba ycen ic coo dina es o 𝑥can be in e p e ed as masses placed a he e ices o 𝜎so 𝑥is he cen e o mass. All hese masses
a e posi i e i and only i 𝑥is inside 𝜎. Fo example, le us conside he 1-simplex 𝜖=⟨𝑣𝑖0, 𝑣𝑖1⟩which is composed o wo e ices o
𝑉. Then |𝜖|is he se o poin s in ℝ𝑛co esponding o he edge wi h endpoin s 𝑣𝑖0and 𝑣𝑖1, and i , o example, 𝑏(𝑥) =(0.5, 0.5) hen
𝑥is he midpoin o |𝜖|.
A simplicial complex 𝐾wi h e ex se 𝑉consis s o a fini e collec ion o simplices sa is ying ha i 𝜎∈𝐾 hen ei he 𝜎=⟨𝑣⟩ o
some 𝑣 ∈𝑉o any ace ( ha is, a nonemp y subse ) o 𝜎is a simplex o 𝐾. Fu he mo e, i 𝜎, 𝜇∈𝐾 hen |𝜎| ∩|𝜇| =∅o |𝜎| ∩|𝜇| =|𝛾|
o some 𝛾∈𝐾. The se ⋃𝜎∈𝐾|𝜎|will be deno ed by |𝐾|. A maximal simplex o 𝐾is a simplex ha is no he ace o any o he
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Fig. 1. On he le , wo iangles ha do no in e sec in a common ace (an edge o a e ex). On he igh , he geome ic ep esen a ion |𝐾|o a pu e 2-simplicial
complex 𝐾composed o h ee maximal 2-simplices ( he iangles 𝜎1, 𝜎2and 𝜎3). The edge 𝜇2is a common ace o 𝜎2and 𝜎3. The edge 𝜇1is a ace o 𝜎1.
simplex o 𝐾. I he maximal simplices o 𝐾a e all 𝑑-simplices hen 𝐾is called a pu e 𝑑-simplicial complex. These concep s a e
illus a ed in Fig. 1.
The ba ycen ic coo dina es o 𝑥wi h espec o he simplicial complex |𝐾|a e defined as he ba ycen ic coo dina es o 𝑥wi h
espec o 𝜎∈𝐾such ha 𝑥 ∈|𝜎|. Le us obse e ha he ba ycen ic coo dina es o 𝑥 ∈|𝐾|a e unique.
An example o simplicial complexes is he Delaunay iangula ion Del(𝑉)defined om he Vo onoi diag am o a gi en fini e se
o e ices 𝑉. The ollowing esul ex ac ed om [9, page 48] is jus an al e na i e defini ion o Delaunay iangula ions.
The emp y ball p ope y [9]: Any subse 𝜎o 𝑉is a simplex o Del(𝑉)i and only i |𝜎|has a ci cumsc ibing open ball emp y o
poin s o 𝑉.
2.2. Simplicial maps
Le 𝐾be a pu e 𝑛-simplicial complex and 𝐿a pu e 𝑘-simplicial complex wi h e ex se s 𝑉and 𝑊, espec i ely. The map
𝜑(0) ∶𝑉→𝑊is called a e ex map i i sa isfies ha he se ob ained om
{𝜑(0)(𝑣𝑖0), … , 𝜑(0)(𝑣𝑖𝑑)}a e emo ing duplica ed e ices
is a simplex in 𝐿whene e ⟨𝑣𝑖0, … , 𝑣𝑖𝑑⟩is a simplex in 𝐾. The e ex map 𝜑(0) always induces a con inuous unc ion, called a
simplicial map 𝜑 ∶|𝐾| →|𝐿|, which is defined as ollows. Le 𝑏(𝑥) =(𝑏0(𝑥), … , 𝑏𝑛(𝑥)) be he ba ycen ic coo dina es o 𝑥 ∈|𝐾|wi h
espec o 𝜎=⟨𝑣𝑖0, … , 𝑣𝑖𝑛⟩ ∈𝐾. Then
𝜑(𝑥)= ∑
𝑗∈0,𝑛
𝑏𝑗(𝑥)𝜑(0)(𝑣𝑖𝑗).
Le us obse e ha 𝜑(𝑥) =𝜑(0)(𝑥)i 𝑥 ∈𝑉.
A special kind o simplicial map used o sol e classifica ion asks will be in oduced in he nex subsec ion.
2.3. Simplicial maps o classifica ion asks
Nex , we will show how a simplicial map can be used o sol e a classifica ion p oblem (see [5] o de ails). F om now on, we will
assume ha he inpu da ase is a fini e se o poin s 𝑉in ℝ𝑛 oge he wi h a se o 𝑘labels Λsuch ha each 𝑣 ∈𝑉is agged wi h a
label 𝜆 aken om Λ.
Fi s ly, he in ui ion is ha he space su ounding he da ase is labelled as unknown. Fo his, we add a new label o Λ, called
unknown label, and a one-ho encoding ep esen a ion 𝑊𝑘+1 ⊂ℝ𝑘+1 o hese 𝑘 +1labels being:
𝑊𝑘+1 ={𝓁𝑗=(0,𝑗
…,0,1,0,𝑘−𝑗
…,0) ∶ 𝑗∈1,𝑘},
whe e he one-ho ec o 𝓁𝑗encodes he 𝑗- h label o Λ o 𝑗∈1, 𝑘and whe e 𝓁0encodes he unknown label.
We now conside a con ex poly ope wi h a e ex se 𝑃su ounding he se 𝑉. The poly ope always exis s since 𝑉is fini e.
Nex , we define a map 𝜑(0) ∶𝑉∪𝑃→𝑊𝑘+1 mapping each e ex 𝑣 ∈𝑉 agged wi h label 𝜆 o he one-ho ec o in 𝑊𝑘+1 ha
encodes he label 𝜆. The e ices o 𝑃a e sen o he e ex 𝓁0. Obse e ha 𝜑(0) is a e ex map.
Le 𝐿deno e he simplicial complex wi h e ex se 𝑊𝑘+1 consis ing o only one maximal 𝑘-simplex and le Del(𝑉∪𝑃)deno e
he Delaunay iangula ion compu ed o he se o poin s 𝑉∪𝑃. No e ha | Del(𝑉∪𝑃)| =. The simplicial map 𝜑 ∶→|𝐿|is
induced by he e ex map 𝜑(0) as explained in Subsec ion 2.2.
Rema k 1. The space |𝐿|can be in e p e ed as he disc e e p obabili y dis ibu ion space Ω𝑘+1 wi h 𝑘 +1 a iables.
As an example, in Fig. 2, on he le , we can see a da ase wi h ou poin s 𝑉={𝑏, 𝑐, 𝑘, 𝑑}, labelled ed and blue. The g een poin s
𝑃={𝑎, 𝑒, 𝑔, 𝑓}a e he e ices o a con ex poly ope con aining 𝑉and a e sen by 𝜑(0) o he g een e ex 𝓁0on he igh . The
simplicial complex 𝐾=Del(𝑉∪𝑃)is d awn on he le and consis s o en maximal 2-simplices. On he igh , he simplicial complex
𝐿consis s o one maximal 2-simplex. The do ed a ows illus a e some examples o 𝜑 ∶→|𝐿|.
In o ma ion Sciences 667 (2024) 120474
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E. Paluzo-Hidalgo, R. Gonzalez-Diaz and M.A. Gu ié ez-Na anjo
Fig. 2. Illus a ion o a simplicial map o a classifica ion ask.
2.4. Simplicial map neu al ne wo ks
A ificial neu al ne wo ks can be seen as pa ame ized eal- alued mappings be ween mul idimensional spaces. Such mappings
a e he composi ion o se e al maps (usually many o hem) ha can be s uc u ed in laye s. In [5], he simplicial map 𝜑defined
abo e was ep esen ed as a wo-hidden-laye eed- o wa d neu al ne wo k 𝜑. This kind o a ificial neu al ne wo k is called
simplicial map neu al ne wo k (SMNN).
In he o iginal defini ion [5], he fi s hidden laye o an SMNN compu es he ba ycen ic coo dina es o Del(𝑉∪𝑃). Howe e ,
we will see he e ha i we p ecompu e he ba ycen ic coo dina es, we can simpli y he a chi ec u e o 𝜑as ollows.
As be o e, conside an inpu da ase consis ing o a fini e se 𝑉⊂ℝ𝑛endowed wi h a se o 𝑘labels and a con ex poly ope 
wi h a se o e ices 𝑃su ounding 𝑉. Le Del(𝑉∪𝑃)be he Delaunay iangula ion wi h e ex se
𝑉∪𝑃={𝜔1,…,𝜔
𝛼}⊆ℝ𝑛.
Then, 𝜑(0) ∶𝑉∪𝑃→𝑊𝑘+1 is a e ex map. Le us assume ha , gi en 𝑥 ∈, we p ecompu e he ba ycen ic coo dina es 𝑏(𝑥) =
(𝑏0(𝑥), … , 𝑏𝑛(𝑥)) ∈ℝ𝑛+1 o 𝑥wi h espec o he 𝑛-simplex 𝜎=⟨𝜔𝑖0, … , 𝜔𝑖𝑛⟩ ∈Del(𝑉∪𝑃)such ha 𝑥 ∈|𝜎|, and ha we also
p ecompu e he ec o
𝜉(𝑥)=(𝜉1(𝑥),…,𝜉
𝛼(𝑥)) ∈ ℝ𝛼
sa is ying ha , o 𝑡 ∈1, 𝛼, 𝜉𝑡(𝑥) =𝑏𝑗(𝑥)i 𝑖𝑗=𝑡 o some 𝑗∈0, 𝑛. Le us ema k ha 𝜉(𝑥)should be a column ec o , bu we will
use ow no a ion, o simplici y.
The SMNN 𝜑induced by 𝜑 ha p edic s he ℎ-label o 𝑥, o ℎ ∈0, 𝑘, has he ollowing a chi ec u e:
•The
numbe o neu ons in he inpu laye is 𝛼.
•The
numbe o neu ons in he ou pu laye is 𝑘 +1.
•The
se o weigh s is ep esen ed as a (𝑘 +1) ×𝛼ma ix such ha he 𝑗- h column o is 𝜑(0)(𝜔𝑡) o 𝑡 ∈1, 𝛼.
Then,
𝜑(𝑥)=⋅𝜉(𝑥).
Obse e ha as defined so a , he SMNN weigh s a e p ecompu ed. Fu he mo e, he compu a ion o he ba ycen ic coo dina es
o he poin s a ound 𝑉implies he calcula ion o he con ex poly ope su ounding 𝑉. Finally, he compu a ion o he Delaunay
iangula ion Del(𝑉∪𝑃)is cos ly i 𝑉∪𝑃has many poin s since i s ime complexi y is 𝑂(𝑛 log 𝑛 +𝑛⌈𝑑
2⌉)(see [9, Chap e 4]).
In he nex sec ions, we will p opose some echniques o o e come he SMNN cons uc ion d awbacks while main aining i s
ad an ages. We will see ha one way o o e come he compu a ion o he con ex poly ope is o conside a hype sphe e 𝑆𝑛
ins ead. We will also see how o a oid he use o he a ificially c ea ed unknown label. Fu he mo e, o educe he cos o Delaunay
compu a ion and add ainabili y o 𝜑 o a oid o e fi ing, a subse 𝑈⊂𝑉 will be conside ed. The se 𝑉will be used o ain and
es a map 𝜑(0)
𝑈∶𝑈→ℝ𝑘. Such a map will induce a con inuous unc ion 𝜑𝑈∶𝐵𝑛→|𝐿|which app oxima es 𝜑.
3. The unknown bounda y and he unc ion 𝝋𝑼
In his sec ion, we will see how o compu e a unc ion 𝜑𝑈 ha app oxima es he simplicial map 𝜑and a oids he compu a ion
o he con ex poly ope and he a ificial conside a ion o he unknown label, educing, a he same ime, he compu a ion o he
Delaunay iangula ion used o cons uc SMNNs. The gene al desc ip ion o he me hodology is desc ibed in Algo i hm 1.
Fi s , le us compu e a hype sphe e su ounding 𝑉. One o he simples ways o do ha is o ansla e 𝑉so ha i s cen e o mass
is placed a he o igin 𝑜 ∈ℝ𝑛. Then, he hype sphe e
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E. Paluzo-Hidalgo, R. Gonzalez-Diaz and M.A. Gu ié ez-Na anjo
Fig. 3. An example o he poin 𝑤𝑥compu ed om 𝑥and he (𝑛−1)-simplex 𝜇=⟨𝑢1,𝑢
2⟩∈Γsuch ha 𝑥∈|𝜎| o 𝜎=⟨𝑤𝑥,𝑢
1,𝑢
2⟩.
𝑆𝑛={𝑤∈ℝ𝑛∶||𝑤||=𝑅}
such ha 𝑅 >max{||𝑣|| ∶𝑣 ∈𝑉}sa isfies ha 𝑆𝑛su ounds 𝑉. Second, le us assume ha we ha e selec ed a subse 𝑈=
{𝑢1, … , 𝑢𝑚} ⊆𝑉 (we will compa e diffe en s a egies o selec 𝑈in Sec ion 7) and ha we ha e compu ed Del(𝑈). Then, we
ha e ha
𝑉⊂𝐵
𝑛={𝑥∈ℝ𝑛∶||𝑥||≤𝑅}and 𝑜∈|Del(𝑈)|.
Le us conside he bounda y o Del(𝑈), deno ed as 𝛿Del(𝑈), which consis s o he se o (𝑛 −1)-simplices ha a e aces o exac ly
one maximal simplex o Del(𝑈).
Now, le us define 𝜉𝑈(𝑥) ∈ℝ𝑚 o any 𝑥 ∈𝐵𝑛as ollows. Gi en 𝑥 ∈𝐵𝑛, o find he 𝑛-simplex 𝜎=⟨𝜔0, … , 𝜔𝑛⟩such ha 𝑥 ∈|𝜎|,
we ha e o conside wo cases: 𝑥 ∈| Del(𝑈)|and 𝑥 ∉| Del(𝑈)|.
I 𝑥 ∈| Del(𝑈)| hen 𝜎is he 𝑛-simplex in Del(𝑈)such ha 𝑥 ∈|𝜎|. I 𝑥 ∉| Del(𝑈)| hen 𝜎is a new 𝑛-simplex defined by he
e ices o an (𝑛 −1)-simplex o 𝛿Del(𝑈)and a new e ex consis ing o he p ojec ion o 𝑥 o 𝑆𝑛. Specifically, i 𝑥 ∉| Del(𝑈)| hen
𝜎is compu ed in he ollowing way:
1. Conside he se
Γ={𝜇∈𝛿Del(𝑈)∶(𝑁⋅𝑢𝑖0+𝑐)(𝑁⋅𝑥+𝑐)<0}
whe e 𝑁is he ec o no mal o he hype plane con aining 𝜇=⟨𝑢𝑖1, … , 𝑢𝑖𝑛⟩, 𝑐=𝑁⋅𝑢𝑖1, and 𝑢𝑖0∈𝑈such ha ⟨𝑢𝑖0, 𝑢𝑖1, … , 𝑢𝑖𝑛⟩ ∈
Del(𝑈).
2. Compu e he poin 𝑤𝑥=𝑅𝑥
||𝑥||∈𝑆𝑛.
3. Find 𝜎=⟨𝑤𝑥, 𝑢𝑖1, … , 𝑢𝑖𝑛⟩such ha
𝜇=⟨𝑢𝑖1,…,𝑢
𝑖𝑛⟩∈Γand 𝑥∈|𝜎|.
Obse e ha , by cons uc ion, 𝜇always exis s since | Del(𝑈)|is a con ex poly ope.
Now, le 𝑏(𝑥) =(𝑏0(𝑥), … , 𝑏𝑛(𝑥)) ∈ℝ𝑛+1 be he ba ycen ic coo dina es o 𝑥wi h espec o 𝜎. Then, 𝜉𝑈(𝑥) =(𝜉1(𝑥), … , 𝜉𝑚(𝑥)) is
he poin in ℝ𝑚sa is ying ha , o 𝑡 ∈1, 𝑚,
𝜉𝑡(𝑥)={𝑏𝑗(𝑥)i 𝑢𝑡=𝜔𝑗 o some 𝑗∈0,𝑛,
0o he wise.
Obse e ha 𝜉𝑈(𝑥)always exis s and is unique. An example o poin s 𝑥and 𝑤𝑥and simplex 𝜇is shown in Fig. 3and Example 1.
Le us obse e ha , hanks o he new defini ion o 𝜉𝑈(𝑥) o 𝑥 ∈𝐵𝑛, i we ha e a map 𝜑(0)
𝑈∶𝑈→ℝ𝑘 hen i induces a con inuous
unc ion 𝜑𝑈∶𝐵𝑛→|𝐿|defined o any 𝑥 ∈𝐵𝑛as:
𝜑𝑈(𝑥)=so max
(∑𝑡∈1,𝑚𝜉𝑡(𝑥)𝜑(0)
𝑈(𝑢𝑡))
=so max(𝑈⋅𝜉𝑈(𝑥))
whe e o 𝑧 =(𝑧1, … , 𝑧𝑘) ∈ℝ𝑘,
so max(𝑧)=(𝑒𝑧1
∑ℎ∈1,𝑘𝑒𝑧ℎ
,…,𝑒𝑧𝑘
∑ℎ∈1,𝑘𝑒𝑧ℎ).

In o ma ion Sciences 667 (2024) 120474
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E. Paluzo-Hidalgo, R. Gonzalez-Diaz and M.A. Gu ié ez-Na anjo
Fig. 4. The ela i e posi ions o he e ices 𝑣𝑖 o 𝑖∈1,5and he poin s 𝑥1and 𝑥2o Example 1.
Le us obse e ha , o ob ain a ca ego ical dis ibu ion om 𝜑𝑈(𝑥) ∈ℝ𝑘, we could jus di ide each o i s coo dina es by he o al
sum. Howe e , so max is in oduced he e o ob ain a simplified o mula o he g adien descen algo i hm as shown in Theo em 1.
Example 1. Le us conside
𝑉={𝑣1=(1
2,1
2),𝑣
2=(1
2,1),𝑣
3=(1,1
2),𝑣
4=(1,1)}
wi h label 𝜆1=0 o 𝑣𝑖wi h 𝑖 =1, 2and 𝜆2=1 o 𝑣𝑖wi h 𝑖 =3, 4. Fi s ly, we ansla e 𝑉so ha he cen e o mass o 𝑉is he o igin
𝑜 ∈ℝ2. The ansla ed da ase

𝑉is
{𝑣1=(−1
4,−1
4),𝑣2=(−1
4,1
4),𝑣3=(1
4,−1
4),𝑣4=(1
4,1
4)}.
Le us conside 𝑥1=(3
4,
3
5)and 𝑥2=(3
4,
5
4). Hence, he ansla ed inpu da a is 𝑥1=(0,
−3
20 )and 𝑥2=(0,
1
2).
To simpli y he explana ion o he me hod, conside 𝑈=
𝑉, ha is, 𝑢𝑖=𝑣𝑖 o 𝑖 ∈1, 4. Then, he maximal simplices o Del(𝑈)a e
𝜎1=⟨𝑣1, 𝑣2, 𝑣3⟩and 𝜎1=⟨𝑣2, 𝑣3, 𝑣4⟩
•The ma ix 𝑈is: (1100
0011
)
• Since he ba ycen ic coo dina es o 𝑥1wi h espec o |𝜎1|a e (0.5, 0.3, 0.2) hen 𝑥1is in |𝜎1| ⊂| Del(𝑈)|and 𝜉𝑈(𝑥1) =
(0.3, 0.2, 0, 0.5). Then
𝜑𝑈(𝑥1)=so max(𝑈⋅𝜉𝑈(𝑥1)) = (0.5,0.5).
•On he o he hand, he poin 𝑥2is ou side | Del(𝑈)|. Assuming ha , o example, we ha e fixed 𝑅 =1, we add a new simplex
𝜎3={𝑤𝑥, 𝑣1, 𝑣2}whe e 𝑤𝑥=𝑣5=(0, 1) which is he p ojec ion o 𝑥2 o he hype sphe e o adius 𝑅cen e ed in he o igin.
See Fig. 4. Then, he ba ycen ic coo dina es o 𝑥2wi h espec o 𝜎3a e (0.33, 0.33, 0.33) and hen 𝜉𝑈(𝑥2) =(0, 0.33, 0.33, 0),
concluding ha
𝜑𝑈(𝑥2)=so max(𝑈⋅𝜉𝑈(𝑥2)) = (0.5,0.5).
The pseudocode o compu ing 𝜑𝑈(𝑥)is p o ided in Algo i hm 1.
The ollowing p ope y holds.
Lemma 1 (Con inui y). Le 𝑥 ∈𝐵𝑛. Then,
lim
𝑦→𝑥𝜉𝑈(𝑦)=𝜉𝑈(𝑥).
P oo . I 𝑥 ∈| Del(𝑈)| hen he esul holds due o he con inui y o he ba ycen ic coo dina es ans o ma ion. I 𝑥 ∉| Del(𝑈)|,
since he o igin 𝑜 ∈| Del(𝑈)|, hen ||𝑥|| ≠0. The e o e, o 𝑦close o 𝑥, ||𝑦|| ≠0and 𝑤𝑦=𝑅𝑦
||𝑦||∈ℝ𝑛. Besides,
lim
𝑦→𝑥𝑤𝑦=𝑤𝑥
and he e o e
In o ma ion Sciences 667 (2024) 120474
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E. Paluzo-Hidalgo, R. Gonzalez-Diaz and M.A. Gu ié ez-Na anjo
Algo i hm 1 Pseudocode o compu e 𝜑𝑈(𝑥) o 𝑥 ∈𝐵𝑛and a subse 𝑈o an inpu da ase 𝑉su ounded by a hype sphe e o adius
𝑅.
Inpu : 𝑈⊂𝑉⊂ℝ𝑛labelled using a se o labels Λ ={𝜆1, … , 𝜆𝑘}, a adius 𝑅, and a poin 𝑥 ∈𝐵𝑛.
Ou pu : The alue o 𝜑𝑈(𝑥)
compu e Del(𝑈)
𝑊𝑘∶= {𝓁𝑗∶= (0,
𝑗−1
…, 0, 1, 0,
𝑘−𝑗
…, 0) o 𝑗∈1, 𝑘}
ini an emp y ma ix 𝑈
ini an emp y ec o 𝑥𝑖(𝑥)
o 𝑢 ∈𝑈do
i 𝜆𝑗is he label o 𝑢 hen
add 𝓁𝑗as a column o 𝑈
end i
end o
o 𝜎maximal simplex o Del(𝑈)do
compu e 𝑏(𝑥)
i 𝑏𝑗≥0 o all 𝑗∈0, 𝑛 hen
s op: compu e 𝜉𝑈(𝑥)
end i
end o
i 𝜉𝑈(𝑥)is emp y hen
compu e Γ, 𝑤𝑥, 𝜇and 𝜎
compu e 𝑏(𝑥)wi h espec o 𝜎and 𝜉𝑈(𝑥)
end i
𝜑𝑈(𝑥) ∶= so max(𝑈⋅𝜉𝑈(𝑥))
lim
𝑦→𝑥𝜉𝑈(𝑦)=𝜉𝑈(𝑥),
concluding he p oo . □
Lemma 2 (Consis ence). Le 𝜑(0) be he map defined in Subsec ion 2.3. I 𝑈=𝑉and 𝜑(0)
𝑈(𝑢) =𝜑(0)(𝑢) o all 𝑢 ∈𝑈 hen
a g max 𝜑𝑈(𝑥) = a g max 𝜑(𝑥); o all 𝑥∈Del(𝑉).
P oo . Le us obse e ha i 𝑈=𝑉and 𝜑(0)
𝑈(𝑢) =𝜑(0)(𝑢) o all 𝑢 ∈𝑈 hen, o any 𝑥 ∈Del(𝑉), we ha e ha :
𝜑(𝑥)= ∑
𝑡∈1,𝑚
𝜉𝑡(𝑥)𝜑(0)(𝑢𝑡)= ∑
𝑡∈1,𝑚
𝜉𝑡(𝑥)𝜑(0)
𝑈(𝑢𝑡).
Then, 𝜑𝑈(𝑥) =so max(𝜑(𝑥))and a g max 𝜑𝑈(𝑥) = a gmax𝜑(𝑥).□
One o he keys o ou s udy is he iden ifica ion o he poin s o ℝ𝑛alloca ed inside a gi en simplex, wi h he se o all p obabili y
dis ibu ions wi h 𝑛 +1suppo alues. In his way, he ba ycen ic coo dina es o a poin can be seen as a p obabili y dis ibu ion.
F om his poin o iew, gi en 𝑥 ∈𝐵𝑛, hen 𝜑(𝑥)and 𝜑𝑈(𝑥)a e bo h in he se |𝐿|o p obabili y dis ibu ions wi h 𝑘suppo poin s.
This is why he ca ego ical c oss-en opy loss unc ion will be used o compa e he simila i y be ween 𝜑and 𝜑𝑈. Specifically, o
𝑣 ∈𝑉, is defined as:
(𝜑𝑈,𝜑,𝑣)=− ∑
ℎ∈1,𝑘
𝑦ℎlog(𝑠ℎ),
whe e 𝜑(0)(𝑣) =(𝑦1, … , 𝑦𝑘)and 𝜑𝑈(𝑣) =(𝑠1, … , 𝑠𝑘).
The ollowing lemma es ablishes a specific se 𝑈⊂𝑉 and a unc ion 𝜑𝑈such ha (𝜑𝑈, 𝜑, 𝑣) =0 o all 𝑣 ∈𝑉.
Lemma 3 (-op imum simplicial map). Le 𝑈be a subse o 𝑉sa is ying, o all 𝑢 ∈𝑈, ha :
1. 𝜑(0)
𝑈(𝑢) =𝜑(0)(𝑢), and
2. i 𝑣 ∈𝑉such ha 𝜑(0)(𝑣) ≠𝜑(0)(𝑢)and ⟨𝑣, 𝑢⟩ ∈Del(𝑉) hen 𝑣 ∈𝑈.
Then,
a g max 𝜑𝑈(𝑥) = a g max 𝜑(𝑥); o all 𝑥∈Del(𝑈).
P oo . As p o ed in [6], unde he assump ions s a ed in his lemma, we ha e ha , o all 𝑥 ∈<| Del(𝑈)|:
𝜑(𝑥)= ∑
𝑡∈1,𝑚
𝜉𝑡(𝑥)𝜑(0)
𝑈(𝑢𝑡)
and hen a g max 𝜑𝑈(𝑥) = a g max 𝜑(𝑥).□
In o ma ion Sciences 667 (2024) 120474
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E. Paluzo-Hidalgo, R. Gonzalez-Diaz and M.A. Gu ié ez-Na anjo
Un o una ely, o compu e he subse 𝑈sa is ying he condi ions s a ed in Lemma 3, we need o compu e he en i e iangula ion
Del(𝑉)which is compu a ionally expensi e, as we ha e al eady men ioned abo e.
4. T aining SMNNs
The no el idea o his pape is o lea n he unc ion 𝜑(0)
𝑈 o any gi en 𝑈⊂𝑉, using he g adien descen algo i hm, in o de o
minimize he loss unc ion (𝜑𝑈, 𝜑, 𝑣) o any 𝑣 ∈𝑉. The ollowing esul p o ides an exp ession o he g adien o in e ms o he
unc ions 𝜑𝑈and 𝜑, and he se 𝑉.
Theo em 1. Le 𝑈={𝑢1, … , 𝑢𝑚}be a subse wi h 𝑚elemen s aken om a fini e se o poin s 𝑉∈ℝ𝑛 agged wi h labels aken om a se
o 𝑘labels. Le 𝜑𝑈∶𝐵𝑛→|𝐿|and 𝜑(0) ∶𝑉→𝑊𝑘. Le us conside ha
{𝜑(0)
𝑈(𝑢𝑡)=(𝑝𝑡
1,…,𝑝
𝑡
𝑘)∶ 𝑡∈1,𝑚}
is a se o a iables. Then, o 𝑣 ∈𝑉,
𝜕(𝜑𝑈,𝜑,𝑣)
𝜕𝑝𝑡
𝑗
=(𝑠𝑗−𝑦𝑗)𝜉𝑡(𝑣)
whe e 𝑗∈1, 𝑘, 𝑡 ∈1, 𝑚, 𝜑(0)(𝑣) =(𝑦1, … , 𝑦𝑘)and 𝜑𝑈(𝑣) =(𝑠1, … , 𝑠𝑘).
P oo . We ha e:
𝜕(𝜑𝑈,𝜑,𝑣)
𝜕𝑝𝑡
𝑗
=−
𝜕(∑ℎ∈1,𝑘𝑦ℎlog(𝑠ℎ))
𝜕𝑝𝑡
𝑗
=− ∑
ℎ∈1,𝑘
𝑦ℎ
𝜕log(𝑠ℎ)
𝜕𝑝𝑖
𝑗
=− ∑
ℎ∈1,𝑘
𝑦ℎ
𝜕log(𝑠ℎ)
𝜕𝑧𝑗
𝜕𝑧𝑗
𝜕𝑝𝑖
𝑗
.
Since 𝑠ℎ=𝑒𝑧ℎ
∑𝑡∈1,𝑘𝑒𝑧𝑡 hen
𝜕log(𝑠ℎ)
𝜕𝑧𝑗
=
𝜕log(𝑒𝑧ℎ
∑𝑡∈1,𝑘𝑒𝑧𝑡)
𝜕𝑧𝑗
=𝜕log(𝑒𝑧ℎ)
𝜕𝑧𝑗
−
𝜕log(∑𝑡∈1,𝑘𝑒𝑧𝑡)
𝜕𝑧𝑗
=𝜕𝑧ℎ
𝜕𝑧𝑗
−1
∑𝑡∈1,𝑘𝑒𝑧𝑡∑
𝑡∈1,𝑘
𝜕𝑒𝑧𝑡
𝜕𝑧𝑗
=𝛿ℎ𝑗 −𝑒𝑧𝑗
∑𝑡∈1,𝑘𝑒𝑧𝑡=𝛿ℎ𝑗 −𝑠𝑗.
Besides, since 𝑧𝑗=∑ℎ∈1,𝑚𝜉ℎ(𝑣)𝑝ℎ
𝑗 hen
𝜕𝑧𝑟
𝜕𝑝𝑡
𝑗
=∑
ℎ∈1,𝑚
𝜉ℎ(𝑣)𝜕𝑝ℎ
𝑟
𝜕𝑝𝑡
𝑗
=𝜉𝑡(𝑣).
Finally,
𝜕(𝜓,𝜑,𝑣)
𝜕𝑝𝑡
𝑗
=− ∑
ℎ∈1,𝑘
𝑦ℎ(𝛿ℎ𝑗 −𝑠𝑗)𝜉𝑡(𝑣)
=−𝜉𝑡(𝑣)(∑
ℎ∈1,𝑘
𝑦ℎ𝛿ℎ𝑗 −𝑠𝑗∑
ℎ∈1,𝑘
𝑦ℎ)
=(𝑠𝑗−𝑦𝑗)𝜉𝑡(𝑣).□
Le us now see how we add ainabili y o he SMNN 𝜑𝑈induced by 𝜑𝑈. Le 𝑉be he aining se and le 𝑈be a suppo se
lying in he same space as 𝑉. Fi s , assuming ha 𝑈={𝑢1, … , 𝑢𝑚}has 𝑚elemen s, hen 𝜑𝑈is a mul iclass pe cep on wi h an
inpu laye wi h 𝑚neu ons ha p edic s he ℎ- h label o ℎ ∈1, 𝑘using he o mula:
𝜑𝑈(𝑥)=so max(
⋅𝜉𝑈(𝑥))
In o ma ion Sciences 667 (2024) 120474
9
E. Paluzo-Hidalgo, R. Gonzalez-Diaz and M.A. Gu ié ez-Na anjo
Fig. 5. Two-dimensional syn he ic da ase wi h poin s di ided in o wo classes: blue and yellow. T iangle-shaped poin s belong o he es se and squa e-shaped
poin s belong o he suppo se 𝑈. The diamond-shaped poin is he e ex on he hype sphe e ( he blue ci cum e ence) used o classi y he iangle-shaped poin 𝑣
(su ounded by a small ed ci cum e ence) ou side he iangula ion.
whe e 
=(𝑝𝑡
𝑗)𝑗∈1,𝑘,𝑡∈1.𝑚is a ma ix o weigh s and 𝜉𝑈(𝑥) ∈ℝ𝑚is ob ained om he ba ycen ic coo dina es o 𝑥 ∈𝐵𝑛as in
Sec ion 3. Le us obse e ha
so max (
⋅𝜉𝑈(𝑥))∈|𝐿|.
The idea is o modi y he ini ial alues o
𝜑(0)
𝑈(𝑢𝑡)=(𝑝𝑡
1,…,𝑝
𝑡
𝑘) o 𝑢𝑡∈𝑈and 𝑡∈1,𝑚,
in o de o ob ain new alues o 𝜑𝑈(𝑣) o 𝑣 ∈𝑉in a way ha he e o (𝜑𝑈,𝜑, 𝑣)dec eases. We will do i by a oiding
ecompu ing Del(𝑈)o he ba ycen ic coo dina es (𝑏0(𝑣), … , 𝑏𝑛(𝑣)) o each 𝑣 ∈𝑉du ing he aining p ocess.
In his way, gi en 𝑣 ∈𝑉, i 𝑣 ∈| Del(𝑈)|, we compu e he maximal simplex 𝜎=⟨𝑢𝑖0, … , 𝑢𝑖𝑛⟩ ∈Del(𝑈)such ha 𝑣 ∈|𝜎|and
𝑖ℎ∈1, 𝑚 o ℎ ∈0, 𝑛. I 𝑣 ∉| Del(𝑈)|, we compu e 𝑤 ∈𝑆𝑛and he simplex 𝜎=⟨𝑤, 𝑢𝑖1, … , 𝑢𝑖𝑛⟩such ha 𝑣 ∈|𝜎|and 𝑖ℎ∈1, 𝑚
o ℎ ∈1, 𝑛. Then we compu e he ba ycen ic coo dina es 𝑏(𝑣)o 𝑣wi h espec o 𝜎and he poin 𝜉𝑈(𝑥) =(𝜉1(𝑥), … , 𝜉𝑚(𝑥)) ∈ℝ𝑚
as in Sec ion 3.
Using he g adien descen algo i hm, we upda e he a iables 𝑝𝑡
𝑗 o 𝑗∈1, 𝑘and 𝑡 ∈1, 𝑚as ollows:
𝑝𝑡
𝑗∶= 𝑝𝑡
𝑗−𝜂
𝜕(𝜑𝑈,𝜑, 𝑣)
𝜕𝑝𝑡
𝑗
=𝑝𝑡
𝑗−𝜂(𝑠𝑗−𝑦𝑗)𝜉𝑡(𝑣).
An illus a i e pic u e o he ole o each poin in a simple wo-dimensional bina y classifica ion p oblem is p o ided in Fig. 5.
The pseudocode o he me hod o ain SMNNs using S ochas ic G adien Descen is p o ided in Algo i hm 2.
Algo i hm 2 Pseudocode o he p oposed me hod o ain SMNNs using SGD.
Inpu : Da ase 𝑉⊂ℝ𝑛su ounded by a hype sphe e o adius 𝑅and a model 𝜑𝑈.
Pa ame e : 𝜂>0
Ou pu : The ained model 𝜑𝑈
ini 
, he ma ix o weigh s o 𝜑𝑈
o 𝑣 ∈𝑉do
compu e 𝜉𝑈(𝑣)as in Sec ion 3
o each column 𝑝o do
𝑝 ∶= 𝑝 −𝜂𝜕(𝜑𝑈,𝜑,𝑣)
𝜕𝑝
end o
end o
𝜑𝑈(𝑥) ∶= so max( 
⋅𝜉𝑈(𝑥))
5. Explainabili y
In his sec ion, we p o ide insigh in o he explainabili y capabili y o SMNNs. In he li e a u e, many diffe en app oaches can be
ound o wha is an explana ion o he p edic ion o an AI model. In ou case, explainabili y will be p o ided based on simila i ies and
dissimila i ies o he poin 𝑥 o be explained wi h he poin s co esponding o he e ices o he simplex 𝜎con aining i . Based on