<s>
Isomap	B-Algorithm
is	O
a	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
method	O
.	O
</s>
<s>
Isomap	B-Algorithm
is	O
used	O
for	O
computing	O
a	O
quasi-isometric	O
,	O
low-dimensional	O
embedding	O
of	O
a	O
set	O
of	O
high-dimensional	O
data	O
points	O
.	O
</s>
<s>
The	O
algorithm	O
provides	O
a	O
simple	O
method	O
for	O
estimating	O
the	O
intrinsic	O
geometry	O
of	O
a	O
data	O
manifold	B-Architecture
based	O
on	O
a	O
rough	O
estimate	O
of	O
each	O
data	O
point	O
’s	O
neighbors	O
on	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
Isomap	B-Algorithm
is	O
highly	O
efficient	O
and	O
generally	O
applicable	O
to	O
a	O
broad	O
range	O
of	O
data	O
sources	O
and	O
dimensionalities	O
.	O
</s>
<s>
Isomap	B-Algorithm
is	O
one	O
representative	O
of	O
isometric	O
mapping	O
methods	O
,	O
and	O
extends	O
metric	O
multidimensional	O
scaling	O
(	O
MDS	O
)	O
by	O
incorporating	O
the	O
geodesic	O
distances	O
imposed	O
by	O
a	O
weighted	O
graph	O
.	O
</s>
<s>
Isomap	B-Algorithm
is	O
distinguished	O
by	O
its	O
use	O
of	O
the	O
geodesic	O
distance	O
induced	O
by	O
a	O
neighborhood	O
graph	O
embedded	O
in	O
the	O
classical	O
scaling	O
.	O
</s>
<s>
This	O
is	O
done	O
to	O
incorporate	O
manifold	B-Architecture
structure	O
in	O
the	O
resulting	O
embedding	O
.	O
</s>
<s>
Isomap	B-Algorithm
defines	O
the	O
geodesic	O
distance	O
to	O
be	O
the	O
sum	O
of	O
edge	O
weights	O
along	O
the	O
shortest	O
path	O
between	O
two	O
nodes	O
(	O
computed	O
using	O
Dijkstra	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
for	O
example	O
)	O
.	O
</s>
<s>
A	O
very	O
high-level	O
description	O
of	O
Isomap	B-Algorithm
algorithm	O
is	O
given	O
below	O
.	O
</s>
<s>
LandMark	O
ISOMAP	B-Algorithm
(	O
L-ISOMAP	O
)	O
:	O
Landmark-Isomap	O
is	O
a	O
variant	O
of	O
Isomap	B-Algorithm
which	O
is	O
faster	O
than	O
Isomap	B-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
accuracy	O
of	O
the	O
manifold	B-Architecture
is	O
compromised	O
by	O
a	O
marginal	O
factor	O
.	O
</s>
<s>
C	O
Isomap	B-Algorithm
:	O
C-Isomap	O
involves	O
magnifying	O
the	O
regions	O
of	O
high	O
density	O
and	O
shrink	O
the	O
regions	O
of	O
low	O
density	O
of	O
data	O
points	O
in	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
This	O
step	O
is	O
vulnerable	O
to	O
"	O
short-circuit	O
errors	O
"	O
if	O
k	O
is	O
too	O
large	O
with	O
respect	O
to	O
the	O
manifold	B-Architecture
structure	O
or	O
if	O
noise	O
in	O
the	O
data	O
moves	O
the	O
points	O
slightly	O
off	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
Following	O
the	O
connection	O
between	O
the	O
classical	O
scaling	O
and	O
PCA	B-Application
,	O
metric	O
MDS	O
can	O
be	O
interpreted	O
as	O
kernel	B-Algorithm
PCA	I-Algorithm
.	O
</s>
<s>
In	O
a	O
similar	O
manner	O
,	O
the	O
geodesic	O
distance	O
matrix	O
in	O
Isomap	B-Algorithm
can	O
be	O
viewed	O
as	O
a	O
kernel	B-Algorithm
matrix	O
.	O
</s>
<s>
However	O
,	O
the	O
kernel	B-Algorithm
matrix	O
K	O
is	O
not	O
always	O
positive	B-Algorithm
semidefinite	I-Algorithm
.	O
</s>
