<s>
Diffusion	B-Algorithm
maps	I-Algorithm
is	O
a	O
dimensionality	B-Algorithm
reduction	I-Algorithm
or	O
feature	B-Algorithm
extraction	I-Algorithm
algorithm	O
introduced	O
by	O
Coifman	O
and	O
Lafon	O
which	O
computes	O
a	O
family	O
of	O
embeddings	O
of	O
a	O
data	O
set	O
into	O
Euclidean	O
space	O
(	O
often	O
low-dimensional	O
)	O
whose	O
coordinates	O
can	O
be	O
computed	O
from	O
the	O
eigenvectors	O
and	O
eigenvalues	O
of	O
a	O
diffusion	O
operator	O
on	O
the	O
data	O
.	O
</s>
<s>
Different	O
from	O
linear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
methods	O
such	O
as	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
,	O
diffusion	B-Algorithm
maps	I-Algorithm
are	O
part	O
of	O
the	O
family	O
of	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
methods	O
which	O
focus	O
on	O
discovering	O
the	O
underlying	O
manifold	B-Architecture
that	O
the	O
data	O
has	O
been	O
sampled	O
from	O
.	O
</s>
<s>
By	O
integrating	O
local	O
similarities	O
at	O
different	O
scales	O
,	O
diffusion	B-Algorithm
maps	I-Algorithm
give	O
a	O
global	O
description	O
of	O
the	O
data-set	O
.	O
</s>
<s>
Compared	O
with	O
other	O
methods	O
,	O
the	O
diffusion	B-Algorithm
map	I-Algorithm
algorithm	O
is	O
robust	O
to	O
noise	O
perturbation	O
and	O
computationally	O
inexpensive	O
.	O
</s>
<s>
Following	O
and	O
,	O
diffusion	B-Algorithm
maps	I-Algorithm
can	O
be	O
defined	O
in	O
four	O
steps	O
.	O
</s>
<s>
Diffusion	B-Algorithm
maps	I-Algorithm
exploit	O
the	O
relationship	O
between	O
heat	O
diffusion	O
and	O
random	O
walk	O
Markov	O
chain	O
.	O
</s>
<s>
Usually	O
,	O
this	O
probability	O
is	O
specified	O
in	O
terms	O
of	O
a	O
kernel	B-Algorithm
function	O
of	O
the	O
two	O
points	O
:	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
popular	O
Gaussian	O
kernel	B-Algorithm
:	O
</s>
<s>
The	O
kernel	B-Algorithm
constitutes	O
the	O
prior	O
definition	O
of	O
the	O
local	O
geometry	O
of	O
the	O
data-set	O
.	O
</s>
<s>
Since	O
a	O
given	O
kernel	B-Algorithm
will	O
capture	O
a	O
specific	O
feature	O
of	O
the	O
data	O
set	O
,	O
its	O
choice	O
should	O
be	O
guided	O
by	O
the	O
application	O
that	O
one	O
has	O
in	O
mind	O
.	O
</s>
<s>
This	O
is	O
a	O
major	O
difference	O
with	O
methods	O
such	O
as	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
where	O
correlations	O
between	O
all	O
data	O
points	O
are	O
taken	O
into	O
account	O
at	O
once	O
.	O
</s>
<s>
Given	O
,	O
we	O
can	O
then	O
construct	O
a	O
reversible	O
discrete-time	O
Markov	O
chain	O
on	O
(	O
a	O
process	O
known	O
as	O
the	O
normalized	O
graph	B-Algorithm
Laplacian	I-Algorithm
construction	O
)	O
:	O
</s>
<s>
Although	O
the	O
new	O
normalized	O
kernel	B-Algorithm
does	O
not	O
inherit	O
the	O
symmetric	O
property	O
,	O
it	O
does	O
inherit	O
the	O
positivity-preserving	O
property	O
and	O
gains	O
a	O
conservation	O
property	O
:	O
</s>
<s>
We	O
apply	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
normalization	O
to	O
this	O
new	O
kernel	B-Algorithm
:	O
</s>
<s>
In	O
some	O
applications	O
,	O
the	O
sampling	O
of	O
the	O
data	O
is	O
generally	O
not	O
related	O
to	O
the	O
geometry	O
of	O
the	O
manifold	B-Architecture
we	O
are	O
interested	O
in	O
describing	O
.	O
</s>
<s>
With	O
,	O
it	O
reduces	O
to	O
the	O
classical	O
graph	B-Algorithm
Laplacian	I-Algorithm
normalization	O
.	O
</s>
<s>
The	O
diffusion	B-Algorithm
map	I-Algorithm
is	O
defined	O
as	O
:	O
</s>
<s>
Thus	O
we	O
get	O
the	O
diffusion	B-Algorithm
map	I-Algorithm
from	O
the	O
original	O
data	O
to	O
a	O
k-dimensional	O
space	O
which	O
is	O
embedded	O
in	O
the	O
original	O
space	O
.	O
</s>
<s>
The	O
basic	O
algorithm	O
framework	O
of	O
diffusion	B-Algorithm
map	I-Algorithm
is	O
as	O
:	O
</s>
<s>
Use	O
diffusion	B-Algorithm
map	I-Algorithm
to	O
get	O
the	O
embedding	O
.	O
</s>
<s>
showed	O
how	O
to	O
design	O
a	O
kernel	B-Algorithm
that	O
reproduces	O
the	O
diffusion	O
induced	O
by	O
a	O
Fokker	O
–	O
Planck	O
equation	O
.	O
</s>
<s>
They	O
also	O
explained	O
that	O
,	O
when	O
the	O
data	O
approximate	O
a	O
manifold	B-Architecture
,	O
one	O
can	O
recover	O
the	O
geometry	O
of	O
this	O
manifold	B-Architecture
by	O
computing	O
an	O
approximation	O
of	O
the	O
Laplace	O
–	O
Beltrami	O
operator	O
.	O
</s>
<s>
Since	O
diffusion	B-Algorithm
maps	I-Algorithm
give	O
a	O
global	O
description	O
of	O
the	O
data-set	O
,	O
they	O
can	O
measure	O
the	O
distances	O
between	O
pairs	O
of	O
sample	O
points	O
in	O
the	O
manifold	B-Architecture
in	O
which	O
the	O
data	O
is	O
embedded	O
.	O
</s>
<s>
Applications	O
based	O
on	O
diffusion	B-Algorithm
maps	I-Algorithm
include	O
face	O
recognition	O
,	O
spectral	B-Algorithm
clustering	I-Algorithm
,	O
low	O
dimensional	O
representation	O
of	O
images	O
,	O
image	O
segmentation	O
,	O
3D	O
model	O
segmentation	O
,	O
speaker	O
verification	O
and	O
identification	O
,	O
sampling	O
on	O
manifolds	B-Architecture
,	O
anomaly	O
detection	O
,	O
image	O
inpainting	O
,	O
revealing	O
brain	O
resting	O
state	O
networks	O
organization	O
and	O
so	O
on	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
diffusion	B-Algorithm
maps	I-Algorithm
framework	O
has	O
been	O
productively	O
extended	O
to	O
complex	O
networks	O
,	O
revealing	O
a	O
functional	O
organisation	O
of	O
networks	O
which	O
differs	O
from	O
the	O
purely	O
topological	O
or	O
structural	O
one	O
.	O
</s>
