Hierarchical Clustering with Prototypes:Minimax Linkage
Jacob Bien, Stanford University
Joint work with Robert Tibshirani, Stanford University
August 4, 2010Joint Statistical Meetings
Jacob Bien JSM 2010
Overview
• Hierarchical clustering is a useful tool for understanding how adataset is organized.
• However, for even moderately sized datasets its effectiveness as avisualization aid degrades.
• Our Goal: To develop a new type of hierarchical clustering that ismore interpretable.
Jacob Bien JSM 2010
Agglomerative Hierarchical Clustering
Builds a hierarchy in a “bottom-up” fashion...
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Jacob Bien JSM 2010
Agglomerative Hierarchical Clustering
Builds a hierarchy in a “bottom-up” fashion...
A B
C
D
E
Jacob Bien JSM 2010
Agglomerative Hierarchical Clustering
Builds a hierarchy in a “bottom-up” fashion...
A B
C
D
E
Jacob Bien JSM 2010
Agglomerative Hierarchical Clustering
Builds a hierarchy in a “bottom-up” fashion...
A B
C
D
E
Jacob Bien JSM 2010
Agglomerative Hierarchical Clustering
Builds a hierarchy in a “bottom-up” fashion...
A B
C
D
E
Jacob Bien JSM 2010
Agglomerative Hierarchical Clustering Review
The approach in words:
• Start with each point in its own cluster.
• Identify the closest two clusters and merge them.
• Repeat.
• Ends when all points are in a single cluster.
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D E B A CJacob Bien JSM 2010
Agglomerative Hierarchical Clustering Review
The approach in words:
• Start with each point in its own cluster.
• Identify the closest two clusters and merge them.
• Repeat.
• Ends when all points are in a single cluster.
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D E B A CJacob Bien JSM 2010
Agglomerative Hierarchical Clustering Review
Linkages
• A linkage is a particular choice of distance between clusters.
• Some standard linkages are...
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Centroid (Eisen's "Average Linkage")
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Jacob Bien JSM 2010
Agglomerative Hierarchical Clustering Review
Cutting trees
• To cut a tree at height h means to consider the clustering you wouldget if you stopped the algorithm the first time all clusters were morethan h apart from each other.
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D E B A CJacob Bien JSM 2010
An Example
Grolier Encyclopedia Dataset: (Courtesy of Sam Roweis)
• Given the number of each word appears in each of 30000encyclopedia articles.
• Two words are similar if they tend to co-occur in articles:
d(xi , xj) = 1− xTi xj
||xi || · ||xj ||
(this is 1 - the uncentered correlation)
• Hierarchical clustering on the 15000 most common words in English.
Jacob Bien JSM 2010
An Example
Hierarchical clustering on 15000 words...w
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tent
brin
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mee
tgr
ound
dram
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ted
build
diffi
cult
brin
gen
orm
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wor
ldw
ide
belie
ved
chan
ged
reco
gniz
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yond
else
whe
recl
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emph
asis
inte
rnat
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lna
tions
wid
espr
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basi
sre
duce
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wid
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part
icul
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cont
inue
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ngre
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com
bine
den
tire
held
them
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third
soon
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ght
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trie
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orld
east
ern
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tern
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rope
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tend
ing
peak
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tern
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rock
ies
rock
yex
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uto
pogr
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peak
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atea
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coas
tal
plai
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rang
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ount
ain
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uble
dur
bani
zatio
nag
raria
nin
dust
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indu
stria
lizat
ion
cont
rolli
ngab
senc
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esen
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tere
dle
ads
fact
ors
resu
lting
resu
ltspr
olon
ged
initi
ally
curr
ently
slow
avai
labi
lity
perc
ent
redu
cede
crea
sein
crea
ses
incr
ease
incr
ease
dse
rvin
gde
scen
ded
poss
ibly
spec
tacu
lar
scat
tere
dtr
end
appr
oxim
atel
ym
illio
nsag
om
illio
nre
view
edbr
iefly
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urag
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uffs
dens
ities
expo
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indu
stria
lized
man
ufac
ture
dim
port
ed raw
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sex
port
impo
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ade
expo
rts
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hern
mos
tis
les
shal
low
wat
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seas
med
iterr
anea
nse
abr
ief
part
ially
inte
rrup
ted
rais
ing
cont
ribut
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link
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dyre
ache
sw
este
rnm
ost
nort
heas
tern
urba
nize
dhi
llslo
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ndup
land
sge
olog
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lyflo
ws
navi
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etr
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este
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aine
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land
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cept
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gly
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omin
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none
rain
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Jacob Bien JSM 2010
Problems with Hierarchical Clustering
• Even for moderately sized datasets, the dendrogram’s interpretativepotential degrades.
1 Branches become very densely packed.2 To interpret a cluster, one traditionally looks at the items in the
cluster and tries to guess what they have in common... not verypractical for large clusters.
Jacob Bien JSM 2010
Our Solution
• Develop a form of hierarchical clustering that automatically labels allinterior nodes of the dendrogram (each representing a cluster).
• Having such labels will fix both problems.
• Simple proposal: Label each cluster by the most highlyrepresentative element contained within (the “prototype” for thatcluster).
Jacob Bien JSM 2010
Our Solution
• Want each cluster well-described by a single prototypical example.
• How can we do this?
• Design a linkage that optimizes for clusters well-represented by asingle element!
Jacob Bien JSM 2010
Our Solution
• Want each cluster well-described by a single prototypical example.
• How can we do this?
• Design a linkage that optimizes for clusters well-represented by asingle element!
Jacob Bien JSM 2010
Our Solution
• Want each cluster well-described by a single prototypical example.
• How can we do this?
• Design a linkage that optimizes for clusters well-represented by asingle element!
Jacob Bien JSM 2010
Minimax Linkage: Definition
Define the minimax linkage to be
d(G , H) = minx∈G∪H
[max
x′∈G∪Hd(x, x′)
],
i.e. the radius of the smallest enclosing ball, centered at a point chosenfrom the two clusters.
Minimax
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Ao, S. I., Yip, K., Ng, M., Cheung, D., Fong, P.-Y., Melhado, I. & Sham, P. C. (2005), ‘Clustag:hierarchical clustering and graph methods for selecting tag snps’, Bioinformatics 21(8), 1735-1736.
Jacob Bien JSM 2010
Minimax Linkage: Definition
• Identify the point whose farthest point is closest.• This central element is taken to be the prototype for the newly
formed cluster G ∪ H.
Minimax
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Jacob Bien JSM 2010
Minimax Linkage Prototypes
So now every node has an associated prototype.
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Jacob Bien JSM 2010
Minimax Linkage: Why use it?
PropertyCutting a minimax linkage dendrogram at height h gives a set ofprototypical observations such that every observation in the dataset iswithin h of its cluster’s prototype.
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Prototypes on Dendrogram: A Geometric View
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Prototypes on Dendrogram: A Geometric View
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Prototypes on Dendrogram: A Geometric View
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Prototypes on Dendrogram: A Geometric View
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Prototypes on Dendrogram: A Geometric View
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Prototypes on Dendrogram: A Geometric View
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Jacob Bien JSM 2010
Back to the Words Example
• We perform minimax clustering on the 15000 words.
• We cut the dendrogram so that there are twenty clusters...
• Each leaf shows the prototype of that cluster.
wye
thno
tice
land
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labi
lity
perc
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sein
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ses
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poss
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spec
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appr
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om
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view
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iefly
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dens
ities
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indu
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man
ufac
ture
dim
port
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expo
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sex
port
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ade
expo
rts
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mos
tis
les
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wat
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med
iterr
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abr
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part
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este
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tern
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dhi
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istic
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equa
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igin
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lyne
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hiev
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tend
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nded
popu
lar
times
latte
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hola
rsar
ose
decl
ine
nota
ble
even
tual
lyse
tfo
unde
dfa
mou
sap
pear
edgr
adua
llyea
rlier
emer
genc
esp
read
clas
sica
lin
fluen
ced
incl
uded
flour
ishe
dba
sed
deve
lopm
ents
crea
ted
own
earli
est
cont
empo
rary
follo
wed
rem
aine
din
trod
uced
med
ieva
lth
roug
hout
cont
inue
dem
erge
din
fluen
cebe
ginn
ing
perio
dm
oder
nla
tehi
stor
yce
ntur
ies
cent
ury
0.0
0.2
0.4
0.6
0.8
1.0
Cluster Dendrogram
Hei
ght
Jacob Bien JSM 2010
Back to the Words Example
• We perform minimax clustering on the 15000 words.• We cut the dendrogram so that there are twenty clusters...• Each leaf shows the prototype of that cluster.
sens
e*sh
ape*
resu
lt*sy
stem
*al
fred
*pl
ay*
food
*tr
ue*
time*
spec
ies*
wat
er*
amer
ican
*ta
il*jo
hn* m
usic
*re
turn
ed*
maj
or*
art*
natio
nal*
cent
ury*
0.99
800.
9985
0.99
900.
9995
1.00
00
Uppercut (h = 0.9981)
Hei
ght
Jacob Bien JSM 2010
The Music branch...
The branch is still too large to see all the leaves...
heire
ssni
ccol
oor
pheu
sco
mpo
sech
risto
phgl
uck
vaug
han
carlo
sw
illia
ms
gunt
her
sieg
frie
dap
pear
ance
sre
cita
lsew
ende
but
disc
ogra
phy
barb
erro
ssin
ite
atro
mod
est
mus
sorg
sky
colla
bora
ted
drew
lyric
ism
ales
sand
rodo
men
ico
fals
taff
aida
verd
ica
sts
plot
sgr
out
idio
mop
erat
icar
ias
oper
aop
eras
doni
zetti
luci
abe
lno
rma
rose
nbe
love
dfo
rbes
ludw
igca
spar
beet
hove
nbo
nnvi
ola
viol
ins
viol
invi
olin
ist
conc
erto
conc
erto
sro
bbin
sce
llo trio
sona
taso
nata
ssu
itequ
arte
tst
ring
hayd
nqu
arte
tsbr
uno
atte
ndin
ggu
stav
mah
ler
juill
iard
schu
bert
sym
phon
icbr
uckn
erno
ssy
mph
onie
ssy
mph
ony
bern
stei
nle
onar
dar
turo
tosc
anin
ior
ches
tra
orch
estr
asph
ilhar
mon
icco
nduc
ting
cond
ucto
rco
nduc
tors
bob
dyla
nhe
llfie
rypr
okof
iev
serg
eire
quie
mbe
rlioz
hect
orbl
ues
pitc
hes
dura
tion
conv
eype
rfor
mer
nota
tion
note
note
sba
rtok
bela
joha
nnes
virt
uoso
cons
erva
tory
com
posi
tions
pian
ist
brah
ms
pian
ists
pian
oad
rian
clau
dio
mad
rigal
mad
rigal
sra
gtim
esp
onso
rshi
piv
esm
arch
esso
pran
osi
nger
sang
sing
ers
sing
ing
solo
ists
choi
rch
orus
choi
rsch
oral
scho
enbe
rgse
rial
stev
eac
com
pany
mus
icia
nst
ylis
ticco
mpo
sing
conc
erts
song
perf
orm
ance
ense
mbl
eso
lopo
lyph
onic
poly
phon
yco
ncer
tor
ches
tral
acco
mpa
nim
ent
com
pose
rvo
cal
song
sm
usic
ians
inst
rum
enta
lco
mpo
sers
mus
icm
usic
al
0.2
0.4
0.6
0.8
1.0
Music Branch
hier (*, "minimax")d
Hei
ght
Jacob Bien JSM 2010
The Music branch...
The branch is still too large to see all the leaves...
heire
ssni
ccol
oor
pheu
sco
mpo
sech
risto
phgl
uck
vaug
han
carlo
sw
illia
ms
gunt
her
sieg
frie
dap
pear
ance
sre
cita
lsew
ende
but
disc
ogra
phy
barb
erro
ssin
ite
atro
mod
est
mus
sorg
sky
colla
bora
ted
drew
lyric
ism
ales
sand
rodo
men
ico
fals
taff
aida
verd
ica
sts
plot
sgr
out
idio
mop
erat
icar
ias
oper
aop
eras
doni
zetti
luci
abe
lno
rma
rose
nbe
love
dfo
rbes
ludw
igca
spar
beet
hove
nbo
nnvi
ola
viol
ins
viol
invi
olin
ist
conc
erto
conc
erto
sro
bbin
sce
llo trio
sona
taso
nata
ssu
itequ
arte
tst
ring
hayd
nqu
arte
tsbr
uno
atte
ndin
ggu
stav
mah
ler
juill
iard
schu
bert
sym
phon
icbr
uckn
erno
ssy
mph
onie
ssy
mph
ony
bern
stei
nle
onar
dar
turo
tosc
anin
ior
ches
tra
orch
estr
asph
ilhar
mon
icco
nduc
ting
cond
ucto
rco
nduc
tors
bob
dyla
nhe
llfie
rypr
okof
iev
serg
eire
quie
mbe
rlioz
hect
orbl
ues
pitc
hes
dura
tion
conv
eype
rfor
mer
nota
tion
note
note
sba
rtok
bela
joha
nnes
virt
uoso
cons
erva
tory
com
posi
tions
pian
ist
brah
ms
pian
ists
pian
oad
rian
clau
dio
mad
rigal
mad
rigal
sra
gtim
esp
onso
rshi
piv
esm
arch
esso
pran
osi
nger
sang
sing
ers
sing
ing
solo
ists
choi
rch
orus
choi
rsch
oral
scho
enbe
rgse
rial
stev
eac
com
pany
mus
icia
nst
ylis
ticco
mpo
sing
conc
erts
song
perf
orm
ance
ense
mbl
eso
lopo
lyph
onic
poly
phon
yco
ncer
tor
ches
tral
acco
mpa
nim
ent
com
pose
rvo
cal
song
sm
usic
ians
inst
rum
enta
lco
mpo
sers
mus
icm
usic
al
0.2
0.4
0.6
0.8
1.0
Music Branch
hier (*, "minimax")d
Hei
ght
Jacob Bien JSM 2010
The Music branch...
gluc
k*
will
iam
s*
gunt
her*
oper
a*
quar
tets
*
mah
ler*
philh
arm
onic
*
fiery
*
berli
oz*
mus
ic*0.
950.
960.
970.
980.
99
Uppercut (h = 0.9543)
Hei
ght
Jacob Bien JSM 2010
Quartets branch
rose
nbe
love
dfo
rbes
ludw
igca
spar
beet
hove
nbo
nnvi
ola
viol
ins
viol
invi
olin
ist co
ncer
toco
ncer
tos
robb
ins
cello
trio
sona
taso
nata
ssu
itequ
arte
tst
ring
hayd
nqu
arte
ts
0.4
0.5
0.6
0.7
0.8
0.9
Quartets Branch
hier (*, "minimax")d
Hei
ght
Jacob Bien JSM 2010
Theoretical Properties of Minimax Linkage
Fisher and Van Ness (1971): “Admissibility” conditions for a linkage
1 No inversions:
d(G , H) ≥ max{h(G ), h(H)}
Jacob Bien JSM 2010
Theoretical Properties of Minimax Linkage
Fisher and Van Ness (1971): “Admissibility” conditions for a linkage
2 “Monotone admissible:” Invariant to monotone transformations ofdistances
3 “Well-structured k-group admissible”
4 “Point proportion admissible”
5 Reducibility
d(G1 ∪ G2, H) ≥ min{d(G1, H), d(G2, H)}
Jacob Bien JSM 2010
Olivetti Faces Dataset
400 images of human faces (64 by 64 pixels).
500
1000
1500
2000
2500
Hei
ght
Jacob Bien JSM 2010
Olivetti Faces Dataset
400 images of human faces (64 by 64 pixels).
600
800
1000
1200
1400
1600
Hei
ght
Jacob Bien JSM 2010
Future
• We will be releasing an R package soon.
• Interactive Flash tool under development (with Daniel Jacobs).
• Contact: [email protected]
Jacob Bien JSM 2010