<s>
In	O
multivariate	B-General_Concept
statistics	I-General_Concept
,	O
spectral	B-Algorithm
clustering	I-Algorithm
techniques	O
make	O
use	O
of	O
the	O
spectrum	O
(	O
eigenvalues	O
)	O
of	O
the	O
similarity	O
matrix	O
of	O
the	O
data	O
to	O
perform	O
dimensionality	B-Algorithm
reduction	I-Algorithm
before	O
clustering	B-Algorithm
in	O
fewer	O
dimensions	O
.	O
</s>
<s>
In	O
application	O
to	O
image	B-Algorithm
segmentation	I-Algorithm
,	O
spectral	B-Algorithm
clustering	I-Algorithm
is	O
known	O
as	O
segmentation-based	B-Algorithm
object	I-Algorithm
categorization	I-Algorithm
.	O
</s>
<s>
The	O
general	O
approach	O
to	O
spectral	B-Algorithm
clustering	I-Algorithm
is	O
to	O
use	O
a	O
standard	O
clustering	B-Algorithm
method	O
(	O
there	O
are	O
many	O
such	O
methods	O
,	O
k-means	B-Algorithm
is	O
discussed	O
below	O
)	O
on	O
relevant	O
eigenvectors	O
of	O
a	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	O
.	O
</s>
<s>
There	O
are	O
many	O
different	O
ways	O
to	O
define	O
a	O
Laplacian	O
which	O
have	O
different	O
mathematical	O
interpretations	O
,	O
and	O
so	O
the	O
clustering	B-Algorithm
will	O
also	O
have	O
different	O
interpretations	O
.	O
</s>
<s>
Spectral	B-Algorithm
clustering	I-Algorithm
is	O
well	O
known	O
to	O
relate	O
to	O
partitioning	O
of	O
a	O
mass-spring	O
system	O
,	O
where	O
each	O
mass	O
is	O
associated	O
with	O
a	O
data	O
point	O
and	O
each	O
spring	O
stiffness	O
corresponds	O
to	O
a	O
weight	O
of	O
an	O
edge	O
describing	O
a	O
similarity	O
of	O
the	O
two	O
related	O
data	O
points	O
,	O
as	O
in	O
the	O
spring	O
system	O
.	O
</s>
<s>
and	O
A	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
masses	O
that	O
are	O
tightly	O
connected	O
by	O
the	O
springs	O
in	O
the	O
mass-spring	O
system	O
evidently	O
move	O
together	O
from	O
the	O
equilibrium	O
position	O
in	O
low-frequency	O
vibration	O
modes	O
,	O
so	O
that	O
the	O
components	O
of	O
the	O
eigenvectors	O
corresponding	O
to	O
the	O
smallest	O
eigenvalues	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
can	O
be	O
used	O
for	O
meaningful	O
clustering	B-Algorithm
of	O
the	O
masses	O
.	O
</s>
<s>
For	O
example	O
,	O
assuming	O
that	O
all	O
the	O
springs	O
and	O
the	O
masses	O
are	O
identical	O
in	O
the	O
2-dimensional	O
spring	O
system	O
pictured	O
,	O
one	O
would	O
intuitively	O
expect	O
that	O
the	O
loosest	O
connected	O
masses	O
on	O
the	O
right-hand	O
side	O
of	O
the	O
system	O
would	O
move	O
with	O
the	O
largest	O
amplitude	O
and	O
in	O
the	O
opposite	O
direction	O
to	O
the	O
rest	O
of	O
the	O
masses	O
when	O
the	O
system	O
is	O
shaken	O
—	O
and	O
this	O
expectation	O
will	O
be	O
confirmed	O
by	O
analyzing	O
components	O
of	O
the	O
eigenvectors	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
corresponding	O
to	O
the	O
smallest	O
eigenvalues	O
,	O
i.e.	O
,	O
the	O
smallest	O
vibration	O
frequencies	O
.	O
</s>
<s>
The	O
goal	O
of	O
normalization	O
is	O
making	O
the	O
diagonal	O
entries	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
to	O
be	O
all	O
unit	O
,	O
also	O
scaling	O
off-diagonal	O
entries	O
correspondingly	O
.	O
</s>
<s>
A	O
popular	O
normalized	O
spectral	B-Algorithm
clustering	I-Algorithm
technique	O
is	O
the	O
normalized	O
cuts	O
algorithm	O
or	O
Shi	O
–	O
Malik	O
algorithm	O
introduced	O
by	O
Jianbo	O
Shi	O
and	O
Jitendra	O
Malik	O
,	O
commonly	O
used	O
for	O
image	B-Algorithm
segmentation	I-Algorithm
.	O
</s>
<s>
and	O
can	O
also	O
be	O
used	O
for	O
spectral	B-Algorithm
clustering	I-Algorithm
.	O
</s>
<s>
A	O
mathematically	O
equivalent	O
algorithm	O
takes	O
the	O
eigenvector	O
corresponding	O
to	O
the	O
largest	O
eigenvalue	O
of	O
the	O
random	B-Algorithm
walk	I-Algorithm
normalized	I-Algorithm
adjacency	I-Algorithm
matrix	O
.	O
</s>
<s>
Now	O
the	O
analysis	O
is	O
reduced	O
to	O
clustering	B-Algorithm
vectors	O
with	O
components	O
,	O
which	O
may	O
be	O
done	O
in	O
various	O
ways	O
.	O
</s>
<s>
This	O
sign-based	O
approach	O
follows	O
the	O
intuitive	O
explanation	O
of	O
spectral	B-Algorithm
clustering	I-Algorithm
via	O
the	O
mass-spring	O
model	O
—	O
in	O
the	O
low	O
frequency	O
vibration	O
mode	O
that	O
the	O
Fiedler	O
vector	O
represents	O
,	O
one	O
cluster	O
data	O
points	O
identified	O
with	O
mutually	O
strongly	O
connected	O
masses	O
would	O
move	O
together	O
in	O
one	O
direction	O
,	O
while	O
in	O
the	O
complement	O
cluster	O
data	O
points	O
identified	O
with	O
remaining	O
masses	O
would	O
move	O
together	O
in	O
the	O
opposite	O
direction	O
.	O
</s>
<s>
The	O
algorithm	O
can	O
be	O
used	O
for	O
hierarchical	B-Algorithm
clustering	I-Algorithm
by	O
repeatedly	O
partitioning	O
the	O
subsets	O
in	O
the	O
same	O
fashion	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
,	O
any	O
vector	O
clustering	B-Algorithm
technique	O
can	O
be	O
used	O
,	O
e.g.	O
,	O
DBSCAN	B-Algorithm
.	O
</s>
<s>
If	O
the	O
similarity	O
matrix	O
has	O
not	O
already	O
been	O
explicitly	O
constructed	O
,	O
the	O
efficiency	O
of	O
spectral	B-Algorithm
clustering	I-Algorithm
may	O
be	O
improved	O
if	O
the	O
solution	O
to	O
the	O
corresponding	O
eigenvalue	O
problem	O
is	O
performed	O
in	O
a	O
matrix-free	O
fashion	O
(	O
without	O
explicitly	O
manipulating	O
or	O
even	O
computing	O
the	O
similarity	O
matrix	O
)	O
,	O
as	O
in	O
the	O
Lanczos	O
algorithm	O
.	O
</s>
<s>
For	O
large-sized	O
graphs	O
,	O
the	O
second	O
eigenvalue	O
of	O
the	O
(	O
normalized	O
)	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	O
is	O
often	O
ill-conditioned	B-Algorithm
,	O
leading	O
to	O
slow	O
convergence	O
of	O
iterative	O
eigenvalue	O
solvers	O
.	O
</s>
<s>
Preconditioning	O
is	O
a	O
key	O
technology	O
accelerating	O
the	O
convergence	O
,	O
e.g.	O
,	O
in	O
the	O
matrix-free	O
LOBPCG	B-Application
method	O
.	O
</s>
<s>
Spectral	B-Algorithm
clustering	I-Algorithm
has	O
been	O
successfully	O
applied	O
on	O
large	O
graphs	O
by	O
first	O
identifying	O
their	O
community	O
structure	O
,	O
and	O
then	O
clustering	B-Algorithm
communities	O
.	O
</s>
<s>
Spectral	B-Algorithm
clustering	I-Algorithm
is	O
closely	O
related	O
to	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
,	O
and	O
dimension	B-Algorithm
reduction	I-Algorithm
techniques	O
such	O
as	O
locally-linear	O
embedding	O
can	O
be	O
used	O
to	O
reduce	O
errors	O
from	O
noise	O
or	O
outliers	O
.	O
</s>
<s>
No	O
matter	O
the	O
algorithm	O
of	O
the	O
spectral	B-Algorithm
clustering	I-Algorithm
,	O
the	O
two	O
main	O
costly	O
items	O
are	O
the	O
construction	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
and	O
determining	O
its	O
eigenvectors	O
for	O
the	O
spectral	O
embedding	O
.	O
</s>
<s>
The	O
need	O
to	O
construct	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
is	O
common	O
for	O
all	O
distance	O
-	O
or	O
correlation-based	O
clustering	B-Algorithm
methods	O
.	O
</s>
<s>
Computing	O
the	O
eigenvectors	O
is	O
specific	O
to	O
spectral	B-Algorithm
clustering	I-Algorithm
only	O
.	O
</s>
<s>
The	O
graph	B-Algorithm
Laplacian	I-Algorithm
can	O
be	O
and	O
commonly	O
is	O
constructed	O
from	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
construction	O
can	O
be	O
performed	O
matrix-free	O
,	O
i.e.	O
,	O
without	O
explicitly	O
forming	O
the	O
matrix	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
and	O
no	O
AO	O
.	O
</s>
<s>
It	O
can	O
also	O
be	O
performed	O
in-place	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
without	O
increasing	O
the	O
memory	O
footprint	O
.	O
</s>
<s>
Either	O
way	O
,	O
the	O
costs	O
of	O
constructing	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
is	O
essentially	O
determined	O
by	O
the	O
costs	O
of	O
constructing	O
the	O
-by	O
-	O
graph	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
a	O
normalized	O
Laplacian	O
has	O
exactly	O
the	O
same	O
eigenvectors	O
as	O
the	O
normalized	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
but	O
with	O
the	O
order	O
of	O
the	O
eigenvalues	O
reversed	O
.	O
</s>
<s>
Thus	O
,	O
instead	O
of	O
computing	O
the	O
eigenvectors	O
corresponding	O
to	O
the	O
smallest	O
eigenvalues	O
of	O
the	O
normalized	O
Laplacian	O
,	O
one	O
can	O
equivalently	O
compute	O
the	O
eigenvectors	O
corresponding	O
to	O
the	O
largest	O
eigenvalues	O
of	O
the	O
normalized	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
without	O
even	O
talking	O
about	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Naive	O
constructions	O
of	O
the	O
graph	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
e.g.	O
,	O
using	O
the	O
RBF	O
kernel	O
,	O
make	O
it	O
dense	O
,	O
thus	O
requiring	O
memory	O
and	O
AO	O
to	O
determine	O
each	O
of	O
the	O
entries	O
of	O
the	O
matrix	O
.	O
</s>
<s>
Algorithms	O
to	O
construct	O
the	O
graph	O
adjacency	B-Algorithm
matrix	I-Algorithm
as	O
a	O
sparse	B-Algorithm
matrix	I-Algorithm
are	O
typically	O
based	O
on	O
a	O
nearest	B-Algorithm
neighbor	I-Algorithm
search	I-Algorithm
,	O
which	O
estimate	O
or	O
sample	O
a	O
neighborhood	O
of	O
a	O
given	O
data	O
point	O
for	O
nearest	O
neighbors	O
,	O
and	O
compute	O
non-zero	O
entries	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
by	O
comparing	O
only	O
pairs	O
of	O
the	O
neighbors	O
.	O
</s>
<s>
The	O
number	O
of	O
the	O
selected	O
nearest	O
neighbors	O
thus	O
determines	O
the	O
number	O
of	O
non-zero	O
entries	O
,	O
and	O
is	O
often	O
fixed	O
so	O
that	O
the	O
memory	O
footprint	O
of	O
the	O
-by	O
-	O
graph	O
adjacency	B-Algorithm
matrix	I-Algorithm
is	O
only	O
,	O
only	O
sequential	O
arithmetic	O
operations	O
are	O
needed	O
to	O
compute	O
the	O
non-zero	O
entries	O
,	O
and	O
the	O
calculations	O
can	O
be	O
trivially	O
run	O
in	O
parallel	O
.	O
</s>
<s>
The	O
cost	O
of	O
computing	O
the	O
-by	O
-	O
(	O
with	O
)	O
matrix	O
of	O
selected	O
eigenvectors	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
is	O
normally	O
proportional	O
to	O
the	O
cost	O
of	O
multiplication	O
of	O
the	O
-by	O
-	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	O
by	O
a	O
vector	O
,	O
which	O
varies	O
greatly	O
whether	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	O
is	O
dense	O
or	O
sparse	O
.	O
</s>
<s>
The	O
very	O
commonly	O
cited	O
in	O
the	O
literature	O
cost	O
comes	O
from	O
choosing	O
and	O
is	O
clearly	O
misleading	O
,	O
since	O
,	O
e.g.	O
,	O
in	O
a	O
hierarchical	O
spectral	B-Algorithm
clustering	I-Algorithm
as	O
determined	O
by	O
the	O
Fiedler	O
vector	O
.	O
</s>
<s>
In	O
the	O
sparse	O
case	O
of	O
the	O
-by	O
-	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	O
with	O
non-zero	O
entries	O
,	O
the	O
cost	O
of	O
the	O
matrix-vector	O
product	O
and	O
thus	O
of	O
computing	O
the	O
-by	O
-	O
with	O
matrix	O
of	O
selected	O
eigenvectors	O
is	O
,	O
with	O
the	O
memory	O
footprint	O
also	O
only	O
—	O
both	O
are	O
the	O
optimal	O
low	O
bounds	O
of	O
complexity	O
of	O
clustering	B-Algorithm
data	O
points	O
.	O
</s>
<s>
Moreover	O
,	O
matrix-free	O
eigenvalue	O
solvers	O
such	O
as	O
LOBPCG	B-Application
can	O
efficiently	O
run	O
in	O
parallel	O
,	O
e.g.	O
,	O
on	O
multiple	O
GPUs	B-Architecture
with	O
distributed	O
memory	O
,	O
resulting	O
not	O
only	O
in	O
high	O
quality	O
clusters	O
,	O
which	O
spectral	B-Algorithm
clustering	I-Algorithm
is	O
famous	O
for	O
,	O
but	O
also	O
top	O
performance	O
.	O
</s>
<s>
Free	O
software	O
implementing	O
spectral	B-Algorithm
clustering	I-Algorithm
is	O
available	O
in	O
large	O
open	O
source	O
projects	O
like	O
scikit-learn	B-Application
using	O
LOBPCG	B-Application
with	O
multigrid	O
preconditioning	O
or	O
ARPACK	B-Application
,	O
MLlib	O
for	O
pseudo-eigenvector	O
clustering	B-Algorithm
using	O
the	O
power	B-Language
iteration	I-Language
method	O
,	O
and	O
R	B-Language
.	O
</s>
<s>
The	O
ideas	O
behind	O
spectral	B-Algorithm
clustering	I-Algorithm
may	O
not	O
be	O
immediately	O
obvious	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
can	O
be	O
described	O
in	O
the	O
context	O
of	O
kernel	O
clustering	B-Algorithm
methods	O
,	O
which	O
reveals	O
several	O
similarities	O
with	O
other	O
approaches	O
.	O
</s>
<s>
shares	O
the	O
objective	O
function	O
with	O
the	O
spectral	B-Algorithm
clustering	I-Algorithm
problem	O
,	O
which	O
can	O
be	O
optimized	O
directly	O
by	O
multi-level	O
methods	O
.	O
</s>
<s>
In	O
the	O
trivial	O
case	O
of	O
determining	O
connected	O
graph	O
components	O
—	O
the	O
optimal	O
clusters	O
with	O
no	O
edges	O
cut	O
—	O
spectral	B-Algorithm
clustering	I-Algorithm
is	O
also	O
related	O
to	O
a	O
spectral	O
version	O
of	O
DBSCAN	B-Algorithm
clustering	B-Algorithm
that	O
finds	O
density-connected	O
components	O
.	O
</s>
<s>
Ravi	O
Kannan	O
,	O
Santosh	O
Vempala	O
and	O
Adrian	O
Vetta	O
proposed	O
a	O
bicriteria	O
measure	O
to	O
define	O
the	O
quality	O
of	O
a	O
given	O
clustering	B-Algorithm
.	O
</s>
<s>
They	O
said	O
that	O
a	O
clustering	B-Algorithm
was	O
an	O
( α	O
,	O
ε	O
)	O
-clustering	O
if	O
the	O
conductance	O
of	O
each	O
cluster	O
(	O
in	O
the	O
clustering	B-Algorithm
)	O
was	O
at	O
least	O
α	O
and	O
the	O
weight	O
of	O
the	O
inter-cluster	O
edges	O
was	O
at	O
most	O
ε	O
fraction	O
of	O
the	O
total	O
weight	O
of	O
all	O
the	O
edges	O
in	O
the	O
graph	O
.	O
</s>
<s>
Spectral	B-Algorithm
clustering	I-Algorithm
has	O
a	O
long	O
history	O
.	O
</s>
<s>
Spectral	B-Algorithm
clustering	I-Algorithm
as	O
a	O
machine	O
learning	O
method	O
was	O
popularized	O
by	O
Shi	O
&	O
Malik	O
and	O
Ng	O
,	O
Jordan	O
,	O
&	O
Weiss	O
.	O
</s>
<s>
Ideas	O
and	O
network	O
measures	O
related	O
to	O
spectral	B-Algorithm
clustering	I-Algorithm
also	O
play	O
an	O
important	O
role	O
in	O
a	O
number	O
of	O
applications	O
apparently	O
different	O
from	O
clustering	B-Algorithm
problems	O
.	O
</s>
