<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
or	O
dimension	B-Algorithm
reduction	I-Algorithm
,	O
is	O
the	O
transformation	O
of	O
data	O
from	O
a	O
high-dimensional	O
space	O
into	O
a	O
low-dimensional	O
space	O
so	O
that	O
the	O
low-dimensional	O
representation	O
retains	O
some	O
meaningful	O
properties	O
of	O
the	O
original	O
data	O
,	O
ideally	O
close	O
to	O
its	O
intrinsic	B-Algorithm
dimension	I-Algorithm
.	O
</s>
<s>
Working	O
in	O
high-dimensional	O
spaces	O
can	O
be	O
undesirable	O
for	O
many	O
reasons	O
;	O
raw	O
data	O
are	O
often	O
sparse	B-Algorithm
as	O
a	O
consequence	O
of	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
,	O
and	O
analyzing	O
the	O
data	O
is	O
usually	O
computationally	O
intractable	O
(	O
hard	O
to	O
control	O
or	O
deal	O
with	O
)	O
.	O
</s>
<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
is	O
common	O
in	O
fields	O
that	O
deal	O
with	O
large	O
numbers	O
of	O
observations	O
and/or	O
large	O
numbers	O
of	O
variables	O
,	O
such	O
as	O
signal	O
processing	O
,	O
speech	B-Application
recognition	I-Application
,	O
neuroinformatics	O
,	O
and	O
bioinformatics	O
.	O
</s>
<s>
Approaches	O
can	O
also	O
be	O
divided	O
into	O
feature	B-General_Concept
selection	I-General_Concept
and	O
feature	B-Algorithm
extraction	I-Algorithm
.	O
</s>
<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
can	O
be	O
used	O
for	O
noise	O
reduction	O
,	O
data	B-Application
visualization	I-Application
,	O
cluster	B-Algorithm
analysis	I-Algorithm
,	O
or	O
as	O
an	O
intermediate	O
step	O
to	O
facilitate	O
other	O
analyses	O
.	O
</s>
<s>
Feature	B-General_Concept
selection	I-General_Concept
approaches	O
try	O
to	O
find	O
a	O
subset	O
of	O
the	O
input	O
variables	O
(	O
also	O
called	O
features	O
or	O
attributes	O
)	O
.	O
</s>
<s>
information	B-Algorithm
gain	I-Algorithm
)	O
,	O
the	O
wrapper	O
strategy	O
(	O
e.g.	O
</s>
<s>
Data	B-General_Concept
analysis	I-General_Concept
such	O
as	O
regression	O
or	O
classification	B-General_Concept
can	O
be	O
done	O
in	O
the	O
reduced	O
space	O
more	O
accurately	O
than	O
in	O
the	O
original	O
space	O
.	O
</s>
<s>
Feature	O
projection	O
(	O
also	O
called	O
feature	B-Algorithm
extraction	I-Algorithm
)	O
transforms	O
the	O
data	O
from	O
the	O
high-dimensional	O
space	O
to	O
a	O
space	O
of	O
fewer	O
dimensions	O
.	O
</s>
<s>
The	O
data	B-General_Concept
transformation	I-General_Concept
may	O
be	O
linear	O
,	O
as	O
in	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
,	O
but	O
many	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
techniques	O
also	O
exist	O
.	O
</s>
<s>
For	O
multidimensional	O
data	O
,	O
tensor	O
representation	O
can	O
be	O
used	O
in	O
dimensionality	B-Algorithm
reduction	I-Algorithm
through	O
multilinear	O
subspace	O
learning	O
.	O
</s>
<s>
The	O
main	O
linear	O
technique	O
for	O
dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
performs	O
a	O
linear	O
mapping	O
of	O
the	O
data	O
to	O
a	O
lower-dimensional	O
space	O
in	O
such	O
a	O
way	O
that	O
the	O
variance	O
of	O
the	O
data	O
in	O
the	O
low-dimensional	O
representation	O
is	O
maximized	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
covariance	O
(	O
and	O
sometimes	O
the	O
correlation	O
)	O
matrix	B-Architecture
of	O
the	O
data	O
is	O
constructed	O
and	O
the	O
eigenvectors	O
on	O
this	O
matrix	B-Architecture
are	O
computed	O
.	O
</s>
<s>
The	O
eigenvectors	O
that	O
correspond	O
to	O
the	O
largest	O
eigenvalues	O
(	O
the	O
principal	B-Application
components	I-Application
)	O
can	O
now	O
be	O
used	O
to	O
reconstruct	O
a	O
large	O
fraction	O
of	O
the	O
variance	O
of	O
the	O
original	O
data	O
.	O
</s>
<s>
NMF	O
decomposes	O
a	O
non-negative	O
matrix	B-Architecture
to	O
the	O
product	O
of	O
two	O
non-negative	O
ones	O
,	O
which	O
has	O
been	O
a	O
promising	O
tool	O
in	O
fields	O
where	O
only	O
non-negative	O
signals	O
exist	O
,	O
such	O
as	O
astronomy	O
.	O
</s>
<s>
NMF	O
is	O
well	O
known	O
since	O
the	O
multiplicative	O
update	O
rule	O
by	O
Lee	O
&	O
Seung	O
,	O
which	O
has	O
been	O
continuously	O
developed	O
:	O
the	O
inclusion	O
of	O
uncertainties	O
,	O
the	O
consideration	O
of	O
missing	O
data	O
and	O
parallel	O
computation	O
,	O
sequential	O
construction	O
which	O
leads	O
to	O
the	O
stability	O
and	O
linearity	O
of	O
NMF	O
,	O
as	O
well	O
as	O
other	O
updates	O
including	O
handling	O
missing	O
data	O
in	O
digital	B-Algorithm
image	I-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
Principal	B-Application
component	I-Application
analysis	I-Application
can	O
be	O
employed	O
in	O
a	O
nonlinear	O
way	O
by	O
means	O
of	O
the	O
kernel	O
trick	O
.	O
</s>
<s>
The	O
resulting	O
technique	O
is	O
called	O
kernel	B-Algorithm
PCA	I-Algorithm
.	O
</s>
<s>
Other	O
prominent	O
nonlinear	O
techniques	O
include	O
manifold	O
learning	O
techniques	O
such	O
as	O
Isomap	B-Algorithm
,	O
locally	B-Algorithm
linear	I-Algorithm
embedding	I-Algorithm
(	O
LLE	O
)	O
,	O
Hessian	O
LLE	O
,	O
Laplacian	O
eigenmaps	O
,	O
and	O
methods	O
based	O
on	O
tangent	O
space	O
analysis	O
.	O
</s>
<s>
These	O
techniques	O
construct	O
a	O
low-dimensional	O
data	B-Application
representation	I-Application
using	O
a	O
cost	O
function	O
that	O
retains	O
local	O
properties	O
of	O
the	O
data	O
,	O
and	O
can	O
be	O
viewed	O
as	O
defining	O
a	O
graph-based	O
kernel	O
for	O
Kernel	B-Algorithm
PCA	I-Algorithm
.	O
</s>
<s>
The	O
most	O
prominent	O
example	O
of	O
such	O
a	O
technique	O
is	O
maximum	B-Algorithm
variance	I-Algorithm
unfolding	I-Algorithm
(	O
MVU	O
)	O
.	O
</s>
<s>
The	O
central	O
idea	O
of	O
MVU	O
is	O
to	O
exactly	O
preserve	O
all	O
pairwise	O
distances	O
between	O
nearest	B-General_Concept
neighbors	I-General_Concept
(	O
in	O
the	O
inner	O
product	O
space	O
)	O
,	O
while	O
maximizing	O
the	O
distances	O
between	O
points	O
that	O
are	O
not	O
nearest	B-General_Concept
neighbors	I-General_Concept
.	O
</s>
<s>
Important	O
examples	O
of	O
such	O
techniques	O
include	O
:	O
classical	O
multidimensional	O
scaling	O
,	O
which	O
is	O
identical	O
to	O
PCA	O
;	O
Isomap	B-Algorithm
,	O
which	O
uses	O
geodesic	O
distances	O
in	O
the	O
data	O
space	O
;	O
diffusion	B-Algorithm
maps	I-Algorithm
,	O
which	O
use	O
diffusion	O
distances	O
in	O
the	O
data	O
space	O
;	O
t-distributed	B-Algorithm
stochastic	I-Algorithm
neighbor	I-Algorithm
embedding	I-Algorithm
(	O
t-SNE	B-Algorithm
)	O
,	O
which	O
minimizes	O
the	O
divergence	O
between	O
distributions	O
over	O
pairs	O
of	O
points	O
;	O
and	O
curvilinear	O
component	O
analysis	O
.	O
</s>
<s>
A	O
different	O
approach	O
to	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
is	O
through	O
the	O
use	O
of	O
autoencoders	B-Algorithm
,	O
a	O
special	O
kind	O
of	O
feedforward	B-Algorithm
neural	I-Algorithm
networks	I-Algorithm
with	O
a	O
bottle-neck	O
hidden	O
layer	O
.	O
</s>
<s>
The	O
training	O
of	O
deep	O
encoders	O
is	O
typically	O
performed	O
using	O
a	O
greedy	O
layer-wise	O
pre-training	O
(	O
e.g.	O
,	O
using	O
a	O
stack	O
of	O
restricted	B-Algorithm
Boltzmann	I-Algorithm
machines	I-Algorithm
)	O
that	O
is	O
followed	O
by	O
a	O
finetuning	O
stage	O
based	O
on	O
backpropagation	B-Algorithm
.	O
</s>
<s>
Linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
(	O
LDA	O
)	O
is	O
a	O
generalization	O
of	O
Fisher	B-General_Concept
's	I-General_Concept
linear	I-General_Concept
discriminant	I-General_Concept
,	O
a	O
method	O
used	O
in	O
statistics	O
,	O
pattern	O
recognition	O
and	O
machine	O
learning	O
to	O
find	O
a	O
linear	O
combination	O
of	O
features	O
that	O
characterizes	O
or	O
separates	O
two	O
or	O
more	O
classes	O
of	O
objects	O
or	O
events	O
.	O
</s>
<s>
GDA	O
deals	O
with	O
nonlinear	O
discriminant	B-General_Concept
analysis	I-General_Concept
using	O
kernel	O
function	O
operator	O
.	O
</s>
<s>
The	O
underlying	O
theory	O
is	O
close	O
to	O
the	O
support-vector	B-Algorithm
machines	I-Algorithm
(	O
SVM	B-Algorithm
)	O
insofar	O
as	O
the	O
GDA	O
method	O
provides	O
a	O
mapping	O
of	O
the	O
input	O
vectors	O
into	O
high-dimensional	O
feature	O
space	O
.	O
</s>
<s>
Autoencoders	B-Algorithm
can	O
be	O
used	O
to	O
learn	O
nonlinear	O
dimension	B-Algorithm
reduction	I-Algorithm
functions	O
and	O
codings	O
together	O
with	O
an	O
inverse	O
function	O
from	O
the	O
coding	O
to	O
the	O
original	O
representation	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
a	O
nonlinear	B-Algorithm
dimensionality	I-Algorithm
reduction	I-Algorithm
technique	O
useful	O
for	O
visualization	O
of	O
high-dimensional	O
datasets	O
.	O
</s>
<s>
Uniform	O
manifold	O
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>
with	O
number	O
of	O
dimensions	O
more	O
than	O
10	O
)	O
,	O
dimension	B-Algorithm
reduction	I-Algorithm
is	O
usually	O
performed	O
prior	O
to	O
applying	O
a	O
K-nearest	B-General_Concept
neighbors	I-General_Concept
algorithm	I-General_Concept
(	O
k-NN	B-General_Concept
)	O
in	O
order	O
to	O
avoid	O
the	O
effects	O
of	O
the	O
curse	B-Algorithm
of	I-Algorithm
dimensionality	I-Algorithm
.	O
</s>
<s>
Feature	B-Algorithm
extraction	I-Algorithm
and	O
dimension	B-Algorithm
reduction	I-Algorithm
can	O
be	O
combined	O
in	O
one	O
step	O
using	O
principal	B-Application
component	I-Application
analysis	I-Application
(	O
PCA	O
)	O
,	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
(	O
LDA	O
)	O
,	O
canonical	O
correlation	O
analysis	O
(	O
CCA	O
)	O
,	O
or	O
non-negative	O
matrix	B-Architecture
factorization	O
(	O
NMF	O
)	O
techniques	O
as	O
a	O
pre-processing	O
step	O
followed	O
by	O
clustering	O
by	O
K-NN	B-General_Concept
on	O
feature	B-Algorithm
vectors	I-Algorithm
in	O
reduced-dimension	O
space	O
.	O
</s>
<s>
when	O
performing	O
similarity	O
search	O
on	O
live	O
video	O
streams	O
,	O
DNA	O
data	O
or	O
high-dimensional	O
time	O
series	O
)	O
running	O
a	O
fast	O
approximate	O
K-NN	B-General_Concept
search	O
using	O
locality-sensitive	B-Algorithm
hashing	I-Algorithm
,	O
random	B-Architecture
projection	I-Architecture
,	O
"	O
sketches	O
"	O
,	O
or	O
other	O
high-dimensional	O
similarity	O
search	O
techniques	O
from	O
the	O
VLDB	O
conference	O
toolbox	O
might	O
be	O
the	O
only	O
feasible	O
option	O
.	O
</s>
<s>
A	O
dimensionality	B-Algorithm
reduction	I-Algorithm
technique	O
that	O
is	O
sometimes	O
used	O
in	O
neuroscience	O
is	O
maximally	O
informative	O
dimensions	O
,	O
which	O
finds	O
a	O
lower-dimensional	O
representation	O
of	O
a	O
dataset	O
such	O
that	O
as	O
much	O
information	O
as	O
possible	O
about	O
the	O
original	O
data	O
is	O
preserved	O
.	O
</s>
