<s>
In	O
statistics	O
and	O
machine	O
learning	O
,	O
Gaussian	B-General_Concept
process	I-General_Concept
approximation	O
is	O
a	O
computational	O
method	O
that	O
accelerates	O
inference	O
tasks	O
in	O
the	O
context	O
of	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
model	O
,	O
most	O
commonly	O
likelihood	O
evaluation	O
and	O
prediction	O
.	O
</s>
<s>
Many	O
of	O
these	O
approximation	O
methods	O
can	O
be	O
expressed	O
in	O
purely	O
linear	B-Language
algebraic	I-Language
or	O
functional	B-Application
analytic	I-Application
terms	O
as	O
matrix	O
or	O
function	O
approximations	O
.	O
</s>
<s>
In	O
statistical	O
modeling	O
,	O
it	O
is	O
often	O
convenient	O
to	O
assume	O
that	O
,	O
the	O
phenomenon	O
under	O
investigation	O
is	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
indexed	O
by	O
which	O
has	O
mean	O
function	O
and	O
covariance	O
function	O
.	O
</s>
<s>
Gaussian	B-Algorithm
process	I-Algorithm
approximations	I-Algorithm
can	O
often	O
be	O
expressed	O
in	O
terms	O
of	O
assumptions	O
on	O
under	O
which	O
and	O
can	O
be	O
calculated	O
with	O
much	O
lower	O
complexity	O
.	O
</s>
<s>
Although	O
most	O
of	O
these	O
methods	O
were	O
developed	O
independently	O
,	O
most	O
of	O
them	O
can	O
be	O
expressed	O
as	O
special	O
cases	O
of	O
the	O
sparse	O
general	O
Vecchia	B-Algorithm
approximation	I-Algorithm
.	O
</s>
<s>
Some	O
of	O
the	O
prominent	O
approximations	O
in	O
this	O
category	O
include	O
the	O
approach	O
based	O
on	O
the	O
equivalence	O
between	O
Gaussian	B-General_Concept
processes	I-General_Concept
with	O
Matern	O
covariance	O
function	O
and	O
stochastic	O
PDEs	O
,	O
periodic	O
embedding	O
,	O
and	O
Nearest	O
Neighbour	O
Gaussian	B-General_Concept
processes	I-General_Concept
.	O
</s>
<s>
One	O
of	O
the	O
most	O
established	O
methods	O
in	O
this	O
class	O
is	O
the	O
Vecchia	B-Algorithm
approximation	I-Algorithm
and	O
its	O
generalization	O
.	O
</s>
<s>
Several	O
other	O
methods	O
can	O
be	O
expressed	O
in	O
this	O
framework	O
,	O
the	O
Multi-resolution	O
Approximation	O
(	O
MRA	O
)	O
,	O
Nearest	O
Neighbour	O
Gaussian	B-General_Concept
Process	I-General_Concept
,	O
Modified	O
Predictive	O
Process	O
and	O
Full-scale	O
approximation	O
.	O
</s>
<s>
While	O
this	O
approach	O
encompasses	O
many	O
methods	O
,	O
the	O
common	O
assumption	O
underlying	O
them	O
all	O
is	O
the	O
assumption	O
,	O
that	O
,	O
the	O
Gaussian	B-General_Concept
process	I-General_Concept
of	O
interest	O
,	O
is	O
effectively	O
low-rank	O
.	O
</s>
<s>
More	O
generally	O
,	O
on	O
top	O
of	O
selecting	O
,	O
one	O
may	O
also	O
find	O
an	O
matrix	O
and	O
assume	O
that	O
,	O
where	O
are	O
values	O
of	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
possibly	O
independent	O
of	O
.	O
</s>
<s>
Many	O
machine	O
learning	O
methods	O
fall	O
into	O
this	O
category	O
,	O
such	O
as	O
subset-of-regressors	O
(	O
SoR	O
)	O
,	O
relevance	B-General_Concept
vector	I-General_Concept
machine	I-General_Concept
,	O
sparse	O
spectrum	O
Gaussian	B-General_Concept
Process	I-General_Concept
and	O
others	O
and	O
they	O
generally	O
differ	O
in	O
the	O
way	O
they	O
derive	O
and	O
.	O
</s>
<s>
Three	O
major	O
members	O
of	O
this	O
group	O
are	O
the	O
meta-kriging	O
algorithm	O
,	O
the	O
gapfill	O
algorithm	O
and	O
Local	O
Approximate	O
Gaussian	B-General_Concept
Process	I-General_Concept
approach	O
.	O
</s>
<s>
Local	O
Approximate	O
Gaussian	B-General_Concept
Process	I-General_Concept
uses	O
a	O
similar	O
logic	O
but	O
constructs	O
a	O
valid	O
stochastic	O
process	O
based	O
on	O
these	O
neighboring	O
values	O
.	O
</s>
