<s>
Nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
,	O
also	O
known	O
as	O
manifold	B-Architecture
learning	O
,	O
refers	O
to	O
various	O
related	O
techniques	O
that	O
aim	O
to	O
project	O
high-dimensional	O
data	O
onto	O
lower-dimensional	O
latent	B-Algorithm
manifolds	I-Algorithm
,	O
with	O
the	O
goal	O
of	O
either	O
visualizing	O
the	O
data	O
in	O
the	O
low-dimensional	O
space	O
,	O
or	O
learning	O
the	O
mapping	O
(	O
either	O
from	O
the	O
high-dimensional	O
space	O
to	O
the	O
low-dimensional	O
embedding	O
or	O
vice	O
versa	O
)	O
itself	O
.	O
</s>
<s>
The	O
techniques	O
described	O
below	O
can	O
be	O
understood	O
as	O
generalizations	O
of	O
linear	O
decomposition	O
methods	O
used	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
such	O
as	O
singular	O
value	O
decomposition	O
and	O
principal	B-Application
component	I-Application
analysis	I-Application
.	O
</s>
<s>
Each	O
row	O
is	O
a	O
sample	O
on	O
a	O
two-dimensional	B-Architecture
manifold	I-Architecture
in	O
1024-dimensional	O
space	O
(	O
a	O
Hamming	O
space	O
)	O
.	O
</s>
<s>
Nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
will	O
discard	O
the	O
correlated	O
information	O
(	O
the	O
letter	O
'	O
A	O
 '	O
)	O
and	O
recover	O
only	O
the	O
varying	O
information	O
(	O
rotation	O
and	O
scale	O
)	O
.	O
</s>
<s>
The	O
image	O
to	O
the	O
right	O
shows	O
sample	O
images	O
from	O
this	O
dataset	O
(	O
to	O
save	O
space	O
,	O
not	O
all	O
input	O
images	O
are	O
shown	O
)	O
,	O
and	O
a	O
plot	O
of	O
the	O
two-dimensional	O
points	O
that	O
results	O
from	O
using	O
a	O
NLDR	O
algorithm	O
(	O
in	O
this	O
case	O
,	O
Manifold	B-Architecture
Sculpting	O
was	O
used	O
)	O
to	O
reduce	O
the	O
data	O
into	O
just	O
two	O
dimensions	O
.	O
</s>
<s>
By	O
comparison	O
,	O
if	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
which	O
is	O
a	O
linear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
algorithm	O
,	O
is	O
used	O
to	O
reduce	O
this	O
same	O
dataset	O
into	O
two	O
dimensions	O
,	O
the	O
resulting	O
values	O
are	O
not	O
so	O
well	O
organized	O
.	O
</s>
<s>
This	O
demonstrates	O
that	O
the	O
high-dimensional	O
vectors	O
(	O
each	O
representing	O
a	O
letter	O
'	O
A	O
 '	O
)	O
that	O
sample	O
this	O
manifold	B-Architecture
vary	O
in	O
a	O
non-linear	O
manner	O
.	O
</s>
<s>
The	O
images	O
obtained	O
by	O
that	O
camera	O
can	O
be	O
considered	O
to	O
be	O
samples	O
on	O
a	O
manifold	B-Architecture
in	O
high-dimensional	O
space	O
,	O
and	O
the	O
intrinsic	O
variables	O
of	O
that	O
manifold	B-Architecture
will	O
represent	O
the	O
robot	O
's	O
position	O
and	O
orientation	O
.	O
</s>
<s>
Invariant	O
manifolds	B-Architecture
are	O
of	O
general	O
interest	O
for	O
model	O
order	O
reduction	O
in	O
dynamical	O
systems	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
there	O
is	O
an	O
attracting	O
invariant	O
manifold	B-Architecture
in	O
the	O
phase	O
space	O
,	O
nearby	O
trajectories	O
will	O
converge	O
onto	O
it	O
and	O
stay	O
on	O
it	O
indefinitely	O
,	O
rendering	O
it	O
a	O
candidate	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
of	O
the	O
dynamical	O
system	O
.	O
</s>
<s>
While	O
such	O
manifolds	B-Architecture
are	O
not	O
guaranteed	O
to	O
exist	O
in	O
general	O
,	O
the	O
theory	O
of	O
spectral	O
submanifolds	O
(	O
SSM	O
)	O
gives	O
conditions	O
for	O
the	O
existence	O
of	O
unique	O
attracting	O
invariant	O
objects	O
in	O
a	O
broad	O
class	O
of	O
dynamical	O
systems	O
.	O
</s>
<s>
Active	O
research	O
in	O
NLDR	O
seeks	O
to	O
unfold	O
the	O
observation	O
manifolds	B-Architecture
associated	O
with	O
dynamical	O
systems	O
to	O
develop	O
modeling	O
techniques	O
.	O
</s>
<s>
Some	O
of	O
the	O
more	O
prominent	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
techniques	O
are	O
listed	O
below	O
.	O
</s>
<s>
Sammon	B-Algorithm
's	I-Algorithm
mapping	I-Algorithm
is	O
one	O
of	O
the	O
first	O
and	O
most	O
popular	O
NLDR	O
techniques	O
.	O
</s>
<s>
The	O
self-organizing	B-Algorithm
map	I-Algorithm
(	O
SOM	B-Algorithm
,	O
also	O
called	O
Kohonen	B-Algorithm
map	I-Algorithm
)	O
and	O
its	O
probabilistic	O
variant	O
generative	B-Algorithm
topographic	I-Algorithm
mapping	I-Algorithm
(	O
GTM	O
)	O
use	O
a	O
point	O
representation	O
in	O
the	O
embedded	O
space	O
to	O
form	O
a	O
latent	O
variable	O
model	O
based	O
on	O
a	O
non-linear	O
mapping	O
from	O
the	O
embedded	O
space	O
to	O
the	O
high-dimensional	O
space	O
.	O
</s>
<s>
Perhaps	O
the	O
most	O
widely	O
used	O
algorithm	O
for	O
dimensional	O
reduction	O
is	O
kernel	B-Algorithm
PCA	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
known	O
to	O
perform	O
poorly	O
with	O
these	O
kernels	O
on	O
the	O
Swiss	O
roll	O
manifold	B-Architecture
.	O
</s>
<s>
However	O
,	O
one	O
can	O
view	O
certain	O
other	O
methods	O
that	O
perform	O
well	O
in	O
such	O
settings	O
(	O
e.g.	O
,	O
Laplacian	O
Eigenmaps	O
,	O
LLE	O
)	O
as	O
special	O
cases	O
of	O
kernel	B-Algorithm
PCA	I-Algorithm
by	O
constructing	O
a	O
data-dependent	O
kernel	O
matrix	O
.	O
</s>
<s>
Principal	O
curves	O
and	O
manifolds	B-Architecture
give	O
the	O
natural	O
geometric	O
framework	O
for	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
and	O
extend	O
the	O
geometric	O
interpretation	O
of	O
PCA	O
by	O
explicitly	O
constructing	O
an	O
embedded	O
manifold	B-Architecture
,	O
and	O
by	O
encoding	O
using	O
standard	O
geometric	O
projection	O
onto	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
How	O
to	O
define	O
the	O
"	O
simplicity	O
"	O
of	O
the	O
manifold	B-Architecture
is	O
problem-dependent	O
,	O
however	O
,	O
it	O
is	O
commonly	O
measured	O
by	O
the	O
intrinsic	O
dimensionality	O
and/or	O
the	O
smoothness	O
of	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
Usually	O
,	O
the	O
principal	O
manifold	B-Architecture
is	O
defined	O
as	O
a	O
solution	O
to	O
an	O
optimization	O
problem	O
.	O
</s>
<s>
The	O
objective	O
function	O
includes	O
a	O
quality	O
of	O
data	O
approximation	O
and	O
some	O
penalty	O
terms	O
for	O
the	O
bending	O
of	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
popular	O
initial	O
approximations	O
are	O
generated	O
by	O
linear	O
PCA	O
and	O
Kohonen	B-Algorithm
's	O
SOM	B-Algorithm
.	O
</s>
<s>
Laplacian	O
eigenmaps	O
uses	O
spectral	O
techniques	O
to	O
perform	O
dimensionality	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
This	O
technique	O
relies	O
on	O
the	O
basic	O
assumption	O
that	O
the	O
data	O
lies	O
in	O
a	O
low-dimensional	O
manifold	B-Architecture
in	O
a	O
high-dimensional	O
space	O
.	O
</s>
<s>
Such	O
techniques	O
can	O
be	O
applied	O
to	O
other	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
algorithms	O
as	O
well	O
.	O
</s>
<s>
Traditional	O
techniques	O
like	O
principal	B-Application
component	I-Application
analysis	I-Application
do	O
not	O
consider	O
the	O
intrinsic	O
geometry	O
of	O
the	O
data	O
.	O
</s>
<s>
the	O
k-nearest	B-General_Concept
neighbor	I-General_Concept
algorithm	I-General_Concept
)	O
.	O
</s>
<s>
The	O
graph	O
thus	O
generated	O
can	O
be	O
considered	O
as	O
a	O
discrete	O
approximation	O
of	O
the	O
low-dimensional	O
manifold	B-Architecture
in	O
the	O
high-dimensional	O
space	O
.	O
</s>
<s>
Minimization	O
of	O
a	O
cost	O
function	O
based	O
on	O
the	O
graph	O
ensures	O
that	O
points	O
close	O
to	O
each	O
other	O
on	O
the	O
manifold	B-Architecture
are	O
mapped	O
close	O
to	O
each	O
other	O
in	O
the	O
low-dimensional	O
space	O
,	O
preserving	O
local	O
distances	O
.	O
</s>
<s>
The	O
eigenfunctions	O
of	O
the	O
Laplace	O
–	O
Beltrami	O
operator	O
on	O
the	O
manifold	B-Architecture
serve	O
as	O
the	O
embedding	O
dimensions	O
,	O
since	O
under	O
mild	O
conditions	O
this	O
operator	O
has	O
a	O
countable	O
spectrum	O
that	O
is	O
a	O
basis	O
for	O
square	O
integrable	O
functions	O
on	O
the	O
manifold	B-Architecture
(	O
compare	O
to	O
Fourier	O
series	O
on	O
the	O
unit	O
circle	O
manifold	B-Architecture
)	O
.	O
</s>
<s>
Isomap	B-Algorithm
is	O
a	O
combination	O
of	O
the	O
Floyd	B-Algorithm
–	I-Algorithm
Warshall	I-Algorithm
algorithm	I-Algorithm
with	O
classic	O
Multidimensional	O
Scaling	O
.	O
</s>
<s>
Isomap	B-Algorithm
assumes	O
that	O
the	O
pair-wise	O
distances	O
are	O
only	O
known	O
between	O
neighboring	O
points	O
,	O
and	O
uses	O
the	O
Floyd	B-Algorithm
–	I-Algorithm
Warshall	I-Algorithm
algorithm	I-Algorithm
to	O
compute	O
the	O
pair-wise	O
distances	O
between	O
all	O
other	O
points	O
.	O
</s>
<s>
Isomap	B-Algorithm
then	O
uses	O
classic	O
MDS	O
to	O
compute	O
the	O
reduced-dimensional	O
positions	O
of	O
all	O
the	O
points	O
.	O
</s>
<s>
Landmark-Isomap	O
is	O
a	O
variant	O
of	O
this	O
algorithm	O
that	O
uses	O
landmarks	O
to	O
increase	O
speed	O
,	O
at	O
the	O
cost	O
of	O
some	O
accuracy	O
.	O
</s>
<s>
In	O
manifold	B-Architecture
learning	O
,	O
the	O
input	O
data	O
is	O
assumed	O
to	O
be	O
sampled	O
from	O
a	O
low	O
dimensional	O
manifold	B-Architecture
that	O
is	O
embedded	O
inside	O
of	O
a	O
higher-dimensional	O
vector	O
space	O
.	O
</s>
<s>
The	O
main	O
intuition	O
behind	O
MVU	O
is	O
to	O
exploit	O
the	O
local	O
linearity	O
of	O
manifolds	B-Architecture
and	O
create	O
a	O
mapping	O
that	O
preserves	O
local	O
neighbourhoods	O
at	O
every	O
point	O
of	O
the	O
underlying	O
manifold	B-Architecture
.	O
</s>
<s>
Locally-linear	O
Embedding	O
(	O
LLE	O
)	O
was	O
presented	O
at	O
approximately	O
the	O
same	O
time	O
as	O
Isomap	B-Algorithm
.	O
</s>
<s>
It	O
has	O
several	O
advantages	O
over	O
Isomap	B-Algorithm
,	O
including	O
faster	O
optimization	O
when	O
implemented	O
to	O
take	O
advantage	O
of	O
sparse	B-Algorithm
matrix	I-Algorithm
algorithms	O
,	O
and	O
better	O
results	O
with	O
many	O
problems	O
.	O
</s>
<s>
LLE	O
also	O
begins	O
by	O
finding	O
a	O
set	O
of	O
the	O
nearest	B-General_Concept
neighbors	I-General_Concept
of	O
each	O
point	O
.	O
</s>
<s>
Generally	O
the	O
data	O
points	O
are	O
reconstructed	O
from	O
K	B-General_Concept
nearest	I-General_Concept
neighbors	I-General_Concept
,	O
as	O
measured	O
by	O
Euclidean	O
distance	O
.	O
</s>
<s>
Like	O
LLE	O
,	O
Hessian	O
LLE	O
is	O
also	O
based	O
on	O
sparse	B-Algorithm
matrix	I-Algorithm
techniques	O
.	O
</s>
<s>
Unfortunately	O
,	O
it	O
has	O
a	O
very	O
costly	O
computational	O
complexity	O
,	O
so	O
it	O
is	O
not	O
well-suited	O
for	O
heavily	O
sampled	O
manifolds	B-Architecture
.	O
</s>
<s>
LTSA	O
is	O
based	O
on	O
the	O
intuition	O
that	O
when	O
a	O
manifold	B-Architecture
is	O
correctly	O
unfolded	O
,	O
all	O
of	O
the	O
tangent	O
hyperplanes	O
to	O
the	O
manifold	B-Architecture
will	O
become	O
aligned	O
.	O
</s>
<s>
It	O
begins	O
by	O
computing	O
the	O
k-nearest	B-General_Concept
neighbors	I-General_Concept
of	O
every	O
point	O
.	O
</s>
<s>
It	O
computes	O
the	O
tangent	O
space	O
at	O
every	O
point	O
by	O
computing	O
the	O
d-first	O
principal	B-Application
components	I-Application
in	O
each	O
local	O
neighborhood	O
.	O
</s>
<s>
Maximum	B-Algorithm
Variance	I-Algorithm
Unfolding	I-Algorithm
,	O
Isomap	B-Algorithm
and	O
Locally	B-Algorithm
Linear	I-Algorithm
Embedding	I-Algorithm
share	O
a	O
common	O
intuition	O
relying	O
on	O
the	O
notion	O
that	O
if	O
a	O
manifold	B-Architecture
is	O
properly	O
unfolded	O
,	O
then	O
variance	O
over	O
the	O
points	O
is	O
maximized	O
.	O
</s>
<s>
Its	O
initial	O
step	O
,	O
like	O
Isomap	B-Algorithm
and	O
Locally	B-Algorithm
Linear	I-Algorithm
Embedding	I-Algorithm
,	O
is	O
finding	O
the	O
k-nearest	B-General_Concept
neighbors	I-General_Concept
of	O
every	O
point	O
.	O
</s>
<s>
Like	O
Locally	B-Algorithm
Linear	I-Algorithm
Embedding	I-Algorithm
,	O
it	O
has	O
no	O
internal	O
model	O
.	O
</s>
<s>
An	O
autoencoder	B-Algorithm
is	O
a	O
feed-forward	O
neural	B-Architecture
network	I-Architecture
which	O
is	O
trained	O
to	O
approximate	O
the	O
identity	O
function	O
.	O
</s>
<s>
When	O
used	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
purposes	O
,	O
one	O
of	O
the	O
hidden	O
layers	O
in	O
the	O
network	O
is	O
limited	O
to	O
contain	O
only	O
a	O
small	O
number	O
of	O
network	O
units	O
.	O
</s>
<s>
Although	O
the	O
idea	O
of	O
autoencoders	B-Algorithm
is	O
quite	O
old	O
,	O
training	O
of	O
deep	O
autoencoders	B-Algorithm
has	O
only	O
recently	O
become	O
possible	O
through	O
the	O
use	O
of	O
restricted	B-Algorithm
Boltzmann	I-Algorithm
machines	I-Algorithm
and	O
stacked	O
denoising	O
autoencoders	B-Algorithm
.	O
</s>
<s>
Related	O
to	O
autoencoders	B-Algorithm
is	O
the	O
NeuroScale	O
algorithm	O
,	O
which	O
uses	O
stress	O
functions	O
inspired	O
by	O
multidimensional	O
scaling	O
and	O
Sammon	B-Algorithm
mappings	I-Algorithm
(	O
see	O
above	O
)	O
to	O
learn	O
a	O
non-linear	O
mapping	O
from	O
the	O
high-dimensional	O
to	O
the	O
embedded	O
space	O
.	O
</s>
<s>
The	O
mappings	O
in	O
NeuroScale	O
are	O
based	O
on	O
radial	B-Algorithm
basis	I-Algorithm
function	I-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
Gaussian	B-General_Concept
process	I-General_Concept
latent	O
variable	O
models	O
(	O
GPLVM	O
)	O
are	O
probabilistic	O
dimensionality	B-Algorithm
reduction	I-Algorithm
methods	O
that	O
use	O
Gaussian	B-General_Concept
Processes	I-General_Concept
(	O
GPs	O
)	O
to	O
find	O
a	O
lower	O
dimensional	O
non-linear	O
embedding	O
of	O
high	O
dimensional	O
data	O
.	O
</s>
<s>
Like	O
kernel	B-Algorithm
PCA	I-Algorithm
they	O
use	O
a	O
kernel	O
function	O
to	O
form	O
a	O
non	O
linear	O
mapping	O
(	O
in	O
the	O
form	O
of	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
)	O
.	O
</s>
<s>
However	O
,	O
in	O
the	O
GPLVM	O
the	O
mapping	O
is	O
from	O
the	O
embedded(latent )	O
space	O
to	O
the	O
data	O
space	O
(	O
like	O
density	O
networks	O
and	O
GTM	O
)	O
whereas	O
in	O
kernel	B-Algorithm
PCA	I-Algorithm
it	O
is	O
in	O
the	O
opposite	O
direction	O
.	O
</s>
<s>
It	O
was	O
originally	O
proposed	O
for	O
visualization	O
of	O
high	O
dimensional	O
data	O
but	O
has	O
been	O
extended	O
to	O
construct	O
a	O
shared	O
manifold	B-Architecture
model	O
between	O
two	O
observation	O
spaces	O
.	O
</s>
<s>
To	O
capture	O
the	O
coupling	O
effect	O
of	O
the	O
pose	O
and	O
gait	O
manifolds	B-Architecture
in	O
the	O
gait	O
analysis	O
,	O
a	O
multi-layer	O
joint	O
gait-pose	O
manifolds	B-Architecture
was	O
proposed	O
.	O
</s>
<s>
t-distributed	B-Algorithm
stochastic	I-Algorithm
neighbor	I-Algorithm
embedding	I-Algorithm
(	O
t-SNE	B-Algorithm
)	O
is	O
widely	O
used	O
.	O
</s>
<s>
The	O
algorithm	O
finds	O
a	O
configuration	O
of	O
data	O
points	O
on	O
a	O
manifold	B-Architecture
by	O
simulating	O
a	O
multi-particle	O
dynamic	O
system	O
on	O
a	O
closed	O
manifold	B-Architecture
,	O
where	O
data	O
points	O
are	O
mapped	O
to	O
particles	O
and	O
distances	O
(	O
or	O
dissimilarity	O
)	O
between	O
data	O
points	O
represent	O
a	O
repulsive	O
force	O
.	O
</s>
<s>
As	O
the	O
manifold	B-Architecture
gradually	O
grows	O
in	O
size	O
the	O
multi-particle	O
system	O
cools	O
down	O
gradually	O
and	O
converges	O
to	O
a	O
configuration	O
that	O
reflects	O
the	O
distance	O
information	O
of	O
the	O
data	O
points	O
.	O
</s>
<s>
The	O
algorithm	O
firstly	O
used	O
the	O
flat	O
torus	O
as	O
the	O
image	O
manifold	B-Architecture
,	O
then	O
it	O
has	O
been	O
extended	O
(	O
in	O
the	O
software	O
to	O
use	O
other	O
types	O
of	O
closed	O
manifolds	B-Architecture
,	O
like	O
the	O
sphere	O
,	O
projective	O
space	O
,	O
and	O
Klein	O
bottle	O
,	O
as	O
image	O
manifolds	B-Architecture
.	O
</s>
<s>
For	O
the	O
contagion	O
map	O
is	O
equivalent	O
to	O
the	O
Isomap	B-Algorithm
algorithm	O
.	O
</s>
<s>
Curvilinear	O
component	O
analysis	O
(	O
CCA	O
)	O
looks	O
for	O
the	O
configuration	O
of	O
points	O
in	O
the	O
output	O
space	O
that	O
preserves	O
original	O
distances	O
as	O
much	O
as	O
possible	O
while	O
focusing	O
on	O
small	O
distances	O
in	O
the	O
output	O
space	O
(	O
conversely	O
to	O
Sammon	B-Algorithm
's	I-Algorithm
mapping	I-Algorithm
which	O
focus	O
on	O
small	O
distances	O
in	O
original	O
space	O
)	O
.	O
</s>
<s>
CDA	O
trains	O
a	O
self-organizing	O
neural	B-Architecture
network	I-Architecture
to	O
fit	O
the	O
manifold	B-Architecture
and	O
seeks	O
to	O
preserve	O
geodesic	O
distances	O
in	O
its	O
embedding	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
Curvilinear	O
Component	O
Analysis	O
(	O
which	O
extended	O
Sammon	B-Algorithm
's	I-Algorithm
mapping	I-Algorithm
)	O
,	O
but	O
uses	O
geodesic	O
distances	O
instead	O
.	O
</s>
<s>
Diffeomorphic	O
Dimensionality	B-Algorithm
Reduction	I-Algorithm
or	O
Diffeomap	O
learns	O
a	O
smooth	O
diffeomorphic	O
mapping	O
which	O
transports	O
the	O
data	O
onto	O
a	O
lower-dimensional	O
linear	O
subspace	O
.	O
</s>
<s>
Manifold	B-Algorithm
alignment	I-Algorithm
takes	O
advantage	O
of	O
the	O
assumption	O
that	O
disparate	O
data	O
sets	O
produced	O
by	O
similar	O
generating	O
processes	O
will	O
share	O
a	O
similar	O
underlying	O
manifold	B-Architecture
representation	O
.	O
</s>
<s>
By	O
learning	O
projections	O
from	O
each	O
original	O
space	O
to	O
the	O
shared	O
manifold	B-Architecture
,	O
correspondences	O
are	O
recovered	O
and	O
knowledge	O
from	O
one	O
domain	O
can	O
be	O
transferred	O
to	O
another	O
.	O
</s>
<s>
Most	O
manifold	B-Algorithm
alignment	I-Algorithm
techniques	O
consider	O
only	O
two	O
data	O
sets	O
,	O
but	O
the	O
concept	O
extends	O
to	O
arbitrarily	O
many	O
initial	O
data	O
sets	O
.	O
</s>
<s>
Diffusion	B-Algorithm
maps	I-Algorithm
leverages	O
the	O
relationship	O
between	O
heat	O
diffusion	O
and	O
a	O
random	O
walk	O
(	O
Markov	O
Chain	O
)	O
;	O
an	O
analogy	O
is	O
drawn	O
between	O
the	O
diffusion	O
operator	O
on	O
a	O
manifold	B-Architecture
and	O
a	O
Markov	O
transition	O
matrix	O
operating	O
on	O
functions	O
defined	O
on	O
the	O
graph	O
whose	O
nodes	O
were	O
sampled	O
from	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
The	O
underlying	O
assumption	O
of	O
diffusion	B-Algorithm
map	I-Algorithm
is	O
that	O
the	O
high-dimensional	O
data	O
lies	O
on	O
a	O
low-dimensional	O
manifold	B-Architecture
of	O
dimension	O
.	O
</s>
<s>
In	O
the	O
above	O
equation	O
,	O
denotes	O
that	O
is	O
a	O
nearest	B-General_Concept
neighbor	I-General_Concept
of	O
.	O
</s>
<s>
Properly	O
,	O
Geodesic	O
distance	O
should	O
be	O
used	O
to	O
actually	O
measure	O
distances	O
on	O
the	O
manifold	B-Architecture
.	O
</s>
<s>
Since	O
the	O
exact	O
structure	O
of	O
the	O
manifold	B-Architecture
is	O
not	O
available	O
,	O
for	O
the	O
nearest	B-General_Concept
neighbors	I-General_Concept
the	O
geodesic	O
distance	O
is	O
approximated	O
by	O
euclidean	O
distance	O
.	O
</s>
<s>
In	O
order	O
to	O
faithfully	O
represent	O
a	O
Markov	O
matrix	O
,	O
must	O
be	O
normalized	O
by	O
the	O
corresponding	O
degree	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
The	O
major	O
difference	O
between	O
diffusion	B-Algorithm
maps	I-Algorithm
and	O
principal	B-Application
component	I-Application
analysis	I-Application
is	O
that	O
only	O
local	O
features	O
of	O
the	O
data	O
are	O
considered	O
in	O
diffusion	B-Algorithm
maps	I-Algorithm
as	O
opposed	O
to	O
taking	O
correlations	O
of	O
the	O
entire	O
data	O
set	O
.	O
</s>
<s>
Nonlinear	O
PCA	O
(	O
NLPCA	O
)	O
uses	O
backpropagation	B-Algorithm
to	O
train	O
a	O
multi-layer	O
perceptron	O
(	O
MLP	O
)	O
to	O
fit	O
to	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
Data-driven	O
high-dimensional	O
scaling	O
(	O
DD-HDS	O
)	O
is	O
closely	O
related	O
to	O
Sammon	B-Algorithm
's	I-Algorithm
mapping	I-Algorithm
and	O
curvilinear	O
component	O
analysis	O
except	O
that	O
(	O
1	O
)	O
it	O
simultaneously	O
penalizes	O
false	O
neighborhoods	O
and	O
tears	O
by	O
focusing	O
on	O
small	O
distances	O
in	O
both	O
original	O
and	O
output	O
space	O
,	O
and	O
that	O
(	O
2	O
)	O
it	O
accounts	O
for	O
concentration	O
of	O
measure	O
phenomenon	O
by	O
adapting	O
the	O
weighting	O
function	O
to	O
the	O
distance	O
distribution	O
.	O
</s>
<s>
Manifold	B-Architecture
Sculpting	O
uses	O
graduated	B-Algorithm
optimization	I-Algorithm
to	O
find	O
an	O
embedding	O
.	O
</s>
<s>
Like	O
other	O
algorithms	O
,	O
it	O
computes	O
the	O
k-nearest	B-General_Concept
neighbors	I-General_Concept
and	O
tries	O
to	O
seek	O
an	O
embedding	O
that	O
preserves	O
relationships	O
in	O
local	O
neighborhoods	O
.	O
</s>
<s>
It	O
can	O
also	O
be	O
used	O
to	O
refine	O
the	O
results	O
from	O
other	O
manifold	B-Architecture
learning	O
algorithms	O
.	O
</s>
<s>
It	O
struggles	O
to	O
unfold	O
some	O
manifolds	B-Architecture
,	O
however	O
,	O
unless	O
a	O
very	O
slow	O
scaling	O
rate	O
is	O
used	O
.	O
</s>
<s>
Aimed	O
at	O
correcting	O
the	O
distortions	O
caused	O
when	O
Isomap	B-Algorithm
is	O
used	O
to	O
map	O
intrinsically	O
non-convex	O
data	O
,	O
TCIE	O
uses	O
weight	O
least-squares	O
MDS	O
in	O
order	O
to	O
obtain	O
a	O
more	O
accurate	O
mapping	O
.	O
</s>
<s>
Uniform	O
manifold	B-Architecture
approximation	O
and	O
projection	O
(	O
UMAP	O
)	O
is	O
a	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
technique	O
.	O
</s>
<s>
Visually	O
,	O
it	O
is	O
similar	O
to	O
t-SNE	B-Algorithm
,	O
but	O
it	O
assumes	O
that	O
the	O
data	O
is	O
uniformly	O
distributed	O
on	O
a	O
locally	O
connected	O
Riemannian	B-Architecture
manifold	I-Architecture
and	O
that	O
the	O
Riemannian	O
metric	O
is	O
locally	O
constant	O
or	O
approximately	O
locally	O
constant	O
.	O
</s>
<s>
The	O
variations	O
tend	O
to	O
be	O
differences	O
in	O
how	O
the	O
proximity	O
data	O
is	O
computed	O
;	O
for	O
example	O
,	O
isomap	B-Algorithm
,	O
locally	B-Algorithm
linear	I-Algorithm
embeddings	I-Algorithm
,	O
maximum	B-Algorithm
variance	I-Algorithm
unfolding	I-Algorithm
,	O
and	O
Sammon	B-Algorithm
mapping	I-Algorithm
(	O
which	O
is	O
not	O
in	O
fact	O
a	O
mapping	O
)	O
are	O
examples	O
of	O
metric	O
multidimensional	O
scaling	O
methods	O
.	O
</s>
