<s>
Low-rank	B-Algorithm
matrix	I-Algorithm
approximations	I-Algorithm
are	O
essential	O
tools	O
in	O
the	O
application	O
of	O
kernel	B-Algorithm
methods	I-Algorithm
to	O
large-scale	O
learning	O
problems	O
.	O
</s>
<s>
Kernel	B-Algorithm
methods	I-Algorithm
(	O
for	O
instance	O
,	O
support	B-Algorithm
vector	I-Algorithm
machines	I-Algorithm
or	O
Gaussian	B-General_Concept
processes	I-General_Concept
)	O
project	O
data	O
points	O
into	O
a	O
high-dimensional	O
or	O
infinite-dimensional	O
feature	B-Algorithm
space	I-Algorithm
and	O
find	O
the	O
optimal	O
splitting	O
hyperplane	O
.	O
</s>
<s>
In	O
the	O
kernel	B-Algorithm
method	I-Algorithm
the	O
data	O
is	O
represented	O
in	O
a	O
kernel	B-Algorithm
matrix	O
(	O
or	O
,	O
Gram	B-Algorithm
matrix	I-Algorithm
)	O
.	O
</s>
<s>
Many	O
algorithms	O
can	O
solve	O
machine	O
learning	O
problems	O
using	O
the	O
kernel	B-Algorithm
matrix	O
.	O
</s>
<s>
The	O
main	O
problem	O
of	O
kernel	B-Algorithm
method	I-Algorithm
is	O
its	O
high	O
computational	B-General_Concept
cost	I-General_Concept
associated	O
with	O
kernel	B-Algorithm
matrices	O
.	O
</s>
<s>
The	O
cost	O
is	O
at	O
least	O
quadratic	O
in	O
the	O
number	O
of	O
training	O
data	O
points	O
,	O
but	O
most	O
kernel	B-Algorithm
methods	I-Algorithm
include	O
computation	O
of	O
matrix	O
inversion	O
or	O
eigenvalue	O
decomposition	O
and	O
the	O
cost	O
becomes	O
cubic	O
in	O
the	O
number	O
of	O
training	O
data	O
.	O
</s>
<s>
Large	O
training	O
sets	O
cause	O
large	O
storage	B-General_Concept
and	I-General_Concept
computational	I-General_Concept
costs	I-General_Concept
.	O
</s>
<s>
Despite	O
low	O
rank	O
decomposition	O
methods	O
(	O
Cholesky	O
decomposition	O
)	O
reduce	O
this	O
cost	O
,	O
they	O
continue	O
to	O
require	O
computing	O
the	O
kernel	B-Algorithm
matrix	O
.	O
</s>
<s>
One	O
of	O
the	O
approaches	O
to	O
deal	O
with	O
this	O
problem	O
is	O
low-rank	B-Algorithm
matrix	I-Algorithm
approximations	I-Algorithm
.	O
</s>
<s>
The	O
most	O
popular	O
examples	O
of	O
them	O
are	O
Nyström	B-Algorithm
method	I-Algorithm
and	O
the	O
random	O
features	O
.	O
</s>
<s>
Both	O
of	O
them	O
have	O
been	O
successfully	O
applied	O
to	O
efficient	O
kernel	B-Algorithm
learning	O
.	O
</s>
<s>
Kernel	B-Algorithm
methods	I-Algorithm
become	O
unfeasible	O
when	O
the	O
number	O
of	O
points	O
is	O
so	O
large	O
such	O
that	O
the	O
kernel	B-Algorithm
matrix	O
cannot	O
be	O
stored	O
in	O
memory	O
.	O
</s>
<s>
If	O
is	O
the	O
number	O
of	O
training	O
examples	O
,	O
the	O
storage	B-General_Concept
and	I-General_Concept
computational	I-General_Concept
cost	I-General_Concept
required	O
to	O
find	O
the	O
solution	O
of	O
the	O
problem	O
using	O
general	O
kernel	B-Algorithm
method	I-Algorithm
is	O
and	O
respectively	O
.	O
</s>
<s>
This	O
speed-up	O
is	O
achieved	O
by	O
using	O
,	O
instead	O
of	O
the	O
kernel	B-Algorithm
matrix	O
,	O
its	O
approximation	O
of	O
rank	O
.	O
</s>
<s>
An	O
advantage	O
of	O
the	O
method	O
is	O
that	O
it	O
is	O
not	O
necessary	O
to	O
compute	O
or	O
store	O
the	O
whole	O
kernel	B-Algorithm
matrix	O
,	O
but	O
only	O
a	O
submatrix	O
of	O
size	O
.	O
</s>
<s>
The	O
method	O
is	O
named	O
"	O
Nyström	O
approximation	O
"	O
because	O
it	O
can	O
be	O
interpreted	O
as	O
a	O
case	O
of	O
the	O
Nyström	B-Algorithm
method	I-Algorithm
from	O
integral	B-Algorithm
equation	I-Algorithm
theory	O
.	O
</s>
<s>
is	O
a	O
kernel	B-Algorithm
matrix	O
for	O
some	O
kernel	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
If	O
and	O
are	O
matrices	O
with	O
'	O
s	O
and	O
'	O
s	O
in	O
the	O
columns	O
and	O
is	O
a	O
diagonal	B-Algorithm
matrix	O
having	O
singular	O
values	O
on	O
the	O
first	O
-entries	O
on	O
the	O
diagonal	B-Algorithm
(	O
all	O
the	O
other	O
elements	O
of	O
the	O
matrix	O
are	O
zeros	O
)	O
:	O
</s>
<s>
Since	O
are	O
orthogonal	B-Algorithm
matrices	I-Algorithm
,	O
</s>
<s>
(	O
is	O
not	O
necessarily	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
)	O
.	O
</s>
<s>
By	O
the	O
characterization	O
for	O
orthogonal	B-Algorithm
matrix	I-Algorithm
:	O
equality	O
holds	O
.	O
</s>
<s>
For	O
a	O
feature	O
map	O
with	O
associated	O
kernel	B-Algorithm
:	O
equality	O
also	O
follows	O
by	O
replacing	O
by	O
the	O
operator	O
such	O
that	O
,	O
,	O
,	O
and	O
by	O
the	O
operator	O
such	O
that	O
,	O
,	O
.	O
</s>
<s>
Once	O
again	O
,	O
a	O
simple	O
inspection	O
shows	O
that	O
the	O
feature	O
map	O
is	O
only	O
needed	O
in	O
the	O
proof	O
while	O
the	O
end	O
result	O
only	O
depends	O
on	O
computing	O
the	O
kernel	B-Algorithm
function	O
.	O
</s>
<s>
In	O
a	O
vector	O
and	O
kernel	B-Algorithm
notation	O
,	O
the	O
problem	O
of	O
Regularized	O
least	O
squares	O
can	O
be	O
rewritten	O
as	O
:	O
</s>
<s>
Let	O
–	O
samples	O
of	O
data	O
,	O
–	O
a	O
randomized	O
feature	O
map	O
(	O
maps	O
a	O
single	O
vector	O
to	O
a	O
vector	O
of	O
higher	O
dimensionality	O
)	O
so	O
that	O
the	O
inner	O
product	O
between	O
a	O
pair	O
of	O
transformed	O
points	O
approximates	O
their	O
kernel	B-Algorithm
evaluation	O
:	O
</s>
<s>
where	O
is	O
the	O
mapping	O
embedded	O
in	O
the	O
RBF	B-Algorithm
kernel	I-Algorithm
.	O
</s>
<s>
Since	O
is	O
low-dimensional	O
,	O
the	O
input	O
can	O
be	O
easily	O
transformed	O
with	O
,	O
after	O
that	O
different	O
linear	O
learning	O
methods	O
to	O
approximate	O
the	O
answer	O
of	O
the	O
corresponding	O
nonlinear	O
kernel	B-Algorithm
can	O
be	O
applied	O
.	O
</s>
<s>
There	O
are	O
different	O
randomized	O
feature	O
maps	O
to	O
compute	O
the	O
approximations	O
to	O
the	O
RBF	B-Algorithm
kernels	I-Algorithm
.	O
</s>
<s>
Random	O
Fourier	O
features	O
map	O
produces	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
approximation	O
to	O
the	O
feature	O
map	O
.	O
</s>
<s>
The	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
considered	O
to	O
be	O
randomized	O
.	O
</s>
<s>
These	O
random	O
features	O
consists	O
of	O
sinusoids	O
randomly	O
drawn	O
from	O
Fourier	B-Algorithm
transform	I-Algorithm
of	O
the	O
kernel	B-Algorithm
to	O
be	O
approximated	O
,	O
where	O
and	O
are	O
random	O
variables	O
.	O
</s>
<s>
The	O
product	O
of	O
the	O
transformed	O
points	O
will	O
approximate	O
a	O
shift-invariant	O
kernel	B-Algorithm
.	O
</s>
<s>
The	O
approaches	O
for	O
large-scale	O
kernel	B-Algorithm
learning	O
(	O
Nyström	B-Algorithm
method	I-Algorithm
and	O
random	O
features	O
)	O
differs	O
in	O
the	O
fact	O
that	O
the	O
Nyström	B-Algorithm
method	I-Algorithm
uses	O
data	O
dependent	O
basis	O
functions	O
while	O
in	O
random	O
features	O
approach	O
the	O
basis	O
functions	O
are	O
sampled	O
from	O
a	O
distribution	O
independent	O
from	O
the	O
training	O
data	O
.	O
</s>
<s>
This	O
difference	O
leads	O
to	O
an	O
improved	O
analysis	O
for	O
kernel	B-Algorithm
learning	O
approaches	O
based	O
on	O
the	O
Nyström	B-Algorithm
method	I-Algorithm
.	O
</s>
<s>
When	O
there	O
is	O
a	O
large	O
gap	O
in	O
the	O
eigen-spectrum	O
of	O
the	O
kernel	B-Algorithm
matrix	O
,	O
approaches	O
based	O
on	O
the	O
Nyström	B-Algorithm
method	I-Algorithm
can	O
achieve	O
better	O
results	O
than	O
Random	O
Features	O
based	O
approach	O
.	O
</s>
