<s>
Kernel	B-Algorithm
methods	I-Algorithm
are	O
a	O
well-established	O
tool	O
to	O
analyze	O
the	O
relationship	O
between	O
input	O
data	O
and	O
the	O
corresponding	O
output	O
of	O
a	O
function	O
.	O
</s>
<s>
Recent	O
development	O
of	O
kernel	B-Algorithm
methods	I-Algorithm
for	O
functions	O
with	O
vector-valued	O
output	O
is	O
due	O
,	O
at	O
least	O
in	O
part	O
,	O
to	O
interest	O
in	O
simultaneously	O
solving	O
related	O
problems	O
.	O
</s>
<s>
Algorithms	O
of	O
this	O
type	O
include	O
multi-task	B-General_Concept
learning	I-General_Concept
(	O
also	O
called	O
multi-output	O
learning	O
or	O
vector-valued	O
learning	O
)	O
,	O
transfer	B-General_Concept
learning	I-General_Concept
,	O
and	O
co-kriging	O
.	O
</s>
<s>
Multi-label	B-Algorithm
classification	I-Algorithm
can	O
be	O
interpreted	O
as	O
mapping	O
inputs	O
to	O
(	O
binary	O
)	O
coding	O
vectors	O
with	O
length	O
equal	O
to	O
the	O
number	O
of	O
classes	O
.	O
</s>
<s>
In	O
Gaussian	B-General_Concept
processes	I-General_Concept
,	O
kernels	O
are	O
called	O
covariance	O
functions	O
.	O
</s>
<s>
See	O
Bayesian	B-General_Concept
interpretation	I-General_Concept
of	I-General_Concept
regularization	I-General_Concept
for	O
the	O
connection	O
between	O
the	O
two	O
perspectives	O
.	O
</s>
<s>
The	O
history	O
of	O
learning	O
vector-valued	O
functions	O
is	O
closely	O
linked	O
to	O
transfer	B-General_Concept
learning	I-General_Concept
-	O
storing	O
knowledge	O
gained	O
while	O
solving	O
one	O
problem	O
and	O
applying	O
it	O
to	O
a	O
different	O
but	O
related	O
problem	O
.	O
</s>
<s>
The	O
fundamental	O
motivation	O
for	O
transfer	B-General_Concept
learning	I-General_Concept
in	O
the	O
field	O
of	O
machine	O
learning	O
was	O
discussed	O
in	O
a	O
NIPS-95	O
workshop	O
on	O
“	O
Learning	O
to	O
Learn	O
,	O
”	O
which	O
focused	O
on	O
the	O
need	O
for	O
lifelong	O
machine	O
learning	O
methods	O
that	O
retain	O
and	O
reuse	O
previously	O
learned	O
knowledge	O
.	O
</s>
<s>
Research	O
on	O
transfer	B-General_Concept
learning	I-General_Concept
has	O
attracted	O
much	O
attention	O
since	O
1995	O
in	O
different	O
names	O
:	O
learning	O
to	O
learn	O
,	O
lifelong	O
learning	O
,	O
knowledge	O
transfer	O
,	O
inductive	B-General_Concept
transfer	I-General_Concept
,	O
multitask	B-General_Concept
learning	I-General_Concept
,	O
knowledge	O
consolidation	O
,	O
context-sensitive	O
learning	O
,	O
knowledge-based	O
inductive	O
bias	O
,	O
metalearning	O
,	O
and	O
incremental/cumulative	O
learning	O
.	O
</s>
<s>
Interest	O
in	O
learning	O
vector-valued	O
functions	O
was	O
particularly	O
sparked	O
by	O
multitask	B-General_Concept
learning	I-General_Concept
,	O
a	O
framework	O
which	O
tries	O
to	O
learn	O
multiple	O
,	O
possibly	O
different	O
tasks	O
simultaneously	O
.	O
</s>
<s>
Much	O
of	O
the	O
initial	O
research	O
in	O
multitask	B-General_Concept
learning	I-General_Concept
in	O
the	O
machine	O
learning	O
community	O
was	O
algorithmic	O
in	O
nature	O
,	O
and	O
applied	O
to	O
methods	O
such	O
as	O
neural	O
networks	O
,	O
decision	O
trees	O
and	O
-nearest	O
neighbors	O
in	O
the	O
1990s	O
.	O
</s>
<s>
The	O
use	O
of	O
probabilistic	O
models	O
and	O
Gaussian	B-General_Concept
processes	I-General_Concept
was	O
pioneered	O
and	O
largely	O
developed	O
in	O
the	O
context	O
of	O
geostatistics	O
,	O
where	O
prediction	O
over	O
vector-valued	O
output	O
data	O
is	O
known	O
as	O
cokriging	O
.	O
</s>
<s>
The	O
estimator	O
of	O
the	O
vector-valued	O
regularization	O
framework	O
can	O
also	O
be	O
derived	O
from	O
a	O
Bayesian	O
viewpoint	O
using	O
Gaussian	B-General_Concept
process	I-General_Concept
methods	O
in	O
the	O
case	O
of	O
a	O
finite	O
dimensional	O
Reproducing	O
kernel	O
Hilbert	O
space	O
.	O
</s>
<s>
The	O
derivation	O
is	O
similar	O
to	O
the	O
scalar-valued	O
case	O
Bayesian	B-General_Concept
interpretation	I-General_Concept
of	I-General_Concept
regularization	I-General_Concept
.	O
</s>
<s>
The	O
vector-valued	O
function	O
,	O
consisting	O
of	O
outputs	O
,	O
is	O
assumed	O
to	O
follow	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
:	O
</s>
<s>
Note	O
,	O
is	O
the	O
graph	B-Algorithm
laplacian	I-Algorithm
.	O
</s>
<s>
See	O
also	O
:	O
graph	B-Algorithm
kernel	I-Algorithm
.	O
</s>
<s>
If	O
the	O
base	O
process	O
is	O
a	O
Gaussian	B-General_Concept
process	I-General_Concept
,	O
the	O
convolved	O
process	O
is	O
Gaussian	O
as	O
well	O
.	O
</s>
<s>
Process	O
convolutions	O
were	O
introduced	O
for	O
multiple	O
outputs	O
in	O
the	O
machine	O
learning	O
community	O
as	O
"	O
dependent	O
Gaussian	B-General_Concept
processes	I-General_Concept
"	O
.	O
</s>
<s>
Approached	O
from	O
the	O
regularization	O
perspective	O
,	O
parameter	O
tuning	O
is	O
similar	O
to	O
the	O
scalar-valued	O
case	O
and	O
can	O
generally	O
be	O
accomplished	O
with	O
cross	B-Application
validation	I-Application
.	O
</s>
<s>
There	O
are	O
many	O
works	O
related	O
to	O
parameter	O
estimation	O
for	O
Gaussian	B-General_Concept
processes	I-General_Concept
.	O
</s>
<s>
A	O
summary	O
of	O
different	O
methods	O
for	O
reducing	O
computational	O
complexity	O
in	O
multi-output	O
Gaussian	B-General_Concept
processes	I-General_Concept
is	O
presented	O
in	O
.	O
</s>
