<s>
In	O
mathematics	O
and	O
statistics	O
,	O
random	B-Architecture
projection	I-Architecture
is	O
a	O
technique	O
used	O
to	O
reduce	B-Algorithm
the	I-Algorithm
dimensionality	I-Algorithm
of	O
a	O
set	O
of	O
points	O
which	O
lie	O
in	O
Euclidean	O
space	O
.	O
</s>
<s>
Random	B-Architecture
projection	I-Architecture
methods	O
are	O
known	O
for	O
their	O
power	O
,	O
simplicity	O
,	O
and	O
low	O
error	O
rates	O
when	O
compared	O
to	O
other	O
methods	O
.	O
</s>
<s>
According	O
to	O
experimental	O
results	O
,	O
random	B-Architecture
projection	I-Architecture
preserves	O
distances	O
well	O
,	O
but	O
empirical	O
results	O
are	O
sparse	O
.	O
</s>
<s>
They	O
have	O
been	O
applied	O
to	O
many	O
natural	O
language	O
tasks	O
under	O
the	O
name	O
random	B-Algorithm
indexing	I-Algorithm
.	O
</s>
<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
as	O
the	O
name	O
suggests	O
,	O
is	O
reducing	O
the	O
number	O
of	O
random	O
variables	O
using	O
various	O
mathematical	O
methods	O
from	O
statistics	O
and	O
machine	O
learning	O
.	O
</s>
<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
is	O
often	O
used	O
to	O
reduce	O
the	O
problem	O
of	O
managing	O
and	O
manipulating	O
large	O
data	O
sets	O
.	O
</s>
<s>
Dimensionality	B-Algorithm
reduction	I-Algorithm
techniques	O
generally	O
use	O
linear	O
transformations	O
in	O
determining	O
the	O
intrinsic	O
dimensionality	O
of	O
the	O
manifold	O
as	O
well	O
as	O
extracting	O
its	O
principal	O
directions	O
.	O
</s>
<s>
For	O
this	O
purpose	O
there	O
are	O
various	O
related	O
techniques	O
,	O
including	O
:	O
principal	B-Application
component	I-Application
analysis	I-Application
,	O
linear	B-General_Concept
discriminant	I-General_Concept
analysis	I-General_Concept
,	O
canonical	O
correlation	O
analysis	O
,	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
,	O
random	B-Architecture
projection	I-Architecture
,	O
etc	O
.	O
</s>
<s>
Random	B-Architecture
projection	I-Architecture
is	O
a	O
simple	O
and	O
computationally	O
efficient	O
way	O
to	O
reduce	B-Algorithm
the	I-Algorithm
dimensionality	I-Algorithm
of	O
data	O
by	O
trading	O
a	O
controlled	O
amount	O
of	O
error	O
for	O
faster	O
processing	O
times	O
and	O
smaller	O
model	O
sizes	O
.	O
</s>
<s>
The	O
dimensions	O
and	O
distribution	O
of	O
random	B-Architecture
projection	I-Architecture
matrices	O
are	O
controlled	O
so	O
as	O
to	O
approximately	O
preserve	O
the	O
pairwise	O
distances	O
between	O
any	O
two	O
samples	O
of	O
the	O
dataset	O
.	O
</s>
<s>
The	O
core	O
idea	O
behind	O
random	B-Architecture
projection	I-Architecture
is	O
given	O
in	O
the	O
Johnson-Lindenstrauss	O
lemma	O
,	O
which	O
states	O
that	O
if	O
points	O
in	O
a	O
vector	O
space	O
are	O
of	O
sufficiently	O
high	O
dimension	O
,	O
then	O
they	O
may	O
be	O
projected	O
into	O
a	O
suitable	O
lower-dimensional	O
space	O
in	O
a	O
way	O
which	O
approximately	O
preserves	O
the	O
distances	O
between	O
the	O
points	O
.	O
</s>
<s>
In	O
random	B-Architecture
projection	I-Architecture
,	O
the	O
original	O
d-dimensional	O
data	O
is	O
projected	O
to	O
a	O
k-dimensional	O
(	O
k	O
<<	O
d	O
)	O
subspace	O
,	O
using	O
a	O
random	O
-	O
dimensional	O
matrix	O
R	O
whose	O
columns	O
have	O
unit	O
lengths	O
.	O
</s>
<s>
Random	B-Architecture
projection	I-Architecture
is	O
computationally	O
simple	O
:	O
form	O
the	O
random	O
matrix	O
"	O
R	O
"	O
and	O
project	O
the	O
data	O
matrix	O
X	O
onto	O
K	O
dimensions	O
of	O
order	O
.	O
</s>
<s>
The	O
second	O
row	O
is	O
a	O
random	O
unit	O
vector	O
from	O
the	O
space	O
orthogonal	B-Application
to	O
the	O
first	O
row	O
,	O
the	O
third	O
row	O
is	O
a	O
random	O
unit	O
vector	O
from	O
the	O
space	O
orthogonal	B-Application
to	O
the	O
first	O
two	O
rows	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
Spherical	O
symmetry	O
:	O
For	O
any	O
orthogonal	B-Application
matrix	O
,	O
RA	O
and	O
R	O
have	O
the	O
same	O
distribution	O
.	O
</s>
<s>
Orthogonality	B-Application
:	O
The	O
rows	O
of	O
R	O
are	O
orthogonal	B-Application
to	O
each	O
other	O
.	O
</s>
<s>
Random	B-Architecture
projection	I-Architecture
can	O
be	O
further	O
condensed	O
by	O
quantization	O
(	O
discretization	O
)	O
,	O
with	O
1-bit	O
(	O
sign	O
random	B-Architecture
projection	I-Architecture
)	O
or	O
multi-bits	O
.	O
</s>
<s>
There	O
are	O
exponentially	O
large	O
(	O
in	O
dimension	O
n	O
)	O
sets	O
of	O
almost	O
orthogonal	B-Application
vectors	O
(	O
with	O
small	O
value	O
of	O
inner	O
products	O
)	O
in	O
n	O
–	O
dimensional	O
Euclidean	O
space	O
.	O
</s>
<s>
This	O
observation	O
is	O
useful	O
in	O
indexing	B-Data_Structure
of	O
high-dimensional	O
data	O
.	O
</s>
<s>
In	O
high	O
dimensions	O
,	O
exponentially	O
large	O
numbers	O
of	O
randomly	O
and	O
independently	O
chosen	O
vectors	O
from	O
equidistribution	O
on	O
a	O
sphere	O
(	O
and	O
from	O
many	O
other	O
distributions	O
)	O
are	O
almost	O
orthogonal	B-Application
with	O
probability	O
close	O
to	O
one	O
.	O
</s>
