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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Dendrogram

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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0.0

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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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Jacob Bien JSM 2010

Prototypes on Dendrogram: A Geometric View

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Jacob Bien JSM 2010

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.

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stor

yce

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ies

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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

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fred

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food

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ies*

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ican

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maj

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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

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frie

dap

pear

ance

sre

cita

lsew

ende

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ogra

phy

barb

erro

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ite

atro

mod

est

mus

sorg

sky

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bora

ted

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ism

ales

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men

ico

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taff

aida

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ica

sts

plot

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out

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icar

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zetti

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abe

lno

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rose

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ist

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ms

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arch

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ers

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ists

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rial

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pany

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icia

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ticco

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ance

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pose

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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

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frie

dap

pear

ance

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cita

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ende

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phy

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ite

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est

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ism

ales

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men

ico

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sts

plot

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out

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icar

ias

oper

aop

eras

doni

zetti

luci

abe

lno

rma

rose

nbe

love

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ludw

igca

spar

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ola

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viol

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ist

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ms

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oad

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rigal

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rigal

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gtim

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onso

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arch

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osi

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sing

ers

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ing

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ists

choi

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orus

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rsch

oral

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enbe

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rial

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eac

com

pany

mus

icia

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ylis

ticco

mpo

sing

conc

erts

song

perf

orm

ance

ense

mbl

eso

lopo

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onic

poly

phon

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pose

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ians

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enta

lco

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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

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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: jbien@stanford.edu

Jacob Bien JSM 2010