<s>
In	O
the	O
field	O
of	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
,	O
matrix	B-Algorithm
regularization	I-Algorithm
generalizes	O
notions	O
of	O
vector	O
regularization	O
to	O
cases	O
where	O
the	O
object	O
to	O
be	O
learned	O
is	O
a	O
matrix	O
.	O
</s>
<s>
Matrix	B-Algorithm
regularization	I-Algorithm
has	O
applications	O
in	O
matrix	O
completion	O
,	O
multivariate	O
regression	O
,	O
and	O
multi-task	B-General_Concept
learning	I-General_Concept
.	O
</s>
<s>
Ideas	O
of	O
feature	O
and	O
group	O
selection	O
can	O
also	O
be	O
extended	O
to	O
matrices	O
,	O
and	O
these	O
can	O
be	O
generalized	O
to	O
the	O
nonparametric	O
case	O
of	O
multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
where	O
defines	O
the	O
empirical	O
error	O
for	O
a	O
given	O
,	O
and	O
is	O
a	O
matrix	B-Algorithm
regularization	I-Algorithm
penalty	O
.	O
</s>
<s>
The	O
setup	O
for	O
multi-task	B-General_Concept
learning	I-General_Concept
is	O
almost	O
the	O
same	O
as	O
the	O
setup	O
for	O
multivariate	O
regression	O
.	O
</s>
<s>
The	O
role	O
of	O
matrix	B-Algorithm
regularization	I-Algorithm
in	O
this	O
setting	O
can	O
be	O
the	O
same	O
as	O
in	O
multivariate	O
regression	O
,	O
but	O
matrix	O
norms	O
can	O
also	O
be	O
used	O
to	O
couple	O
learning	O
problems	O
across	O
tasks	O
.	O
</s>
<s>
When	O
the	O
relationship	O
between	O
tasks	O
is	O
known	O
to	O
lie	O
on	O
a	O
graph	O
,	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
can	O
be	O
used	O
to	O
couple	O
the	O
learning	O
problems	O
.	O
</s>
<s>
For	O
example	O
,	O
matrix	B-Algorithm
regularization	I-Algorithm
with	O
a	O
Schatten	O
1-norm	O
,	O
also	O
called	O
the	O
nuclear	O
norm	O
,	O
can	O
be	O
used	O
to	O
enforce	O
sparsity	O
in	O
the	O
spectrum	O
of	O
a	O
matrix	O
.	O
</s>
<s>
the	O
Lasso	B-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
norm	O
is	O
used	O
in	O
multi-task	B-General_Concept
learning	I-General_Concept
to	O
group	O
features	O
across	O
tasks	O
,	O
such	O
that	O
all	O
the	O
elements	O
in	O
a	O
given	O
row	O
of	O
the	O
coefficient	O
matrix	O
can	O
be	O
forced	O
to	O
zero	O
as	O
a	O
group	O
.	O
</s>
<s>
Algorithms	O
for	O
solving	O
these	O
group	O
sparsity	O
problems	O
extend	O
the	O
more	O
well-known	O
Lasso	B-Algorithm
and	O
group	O
Lasso	B-Algorithm
methods	O
by	O
allowing	O
overlapping	O
groups	O
,	O
for	O
example	O
,	O
and	O
have	O
been	O
implemented	O
via	O
matching	B-General_Concept
pursuit	I-General_Concept
:	O
and	O
proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
The	O
ideas	O
of	O
structured	O
sparsity	O
and	O
feature	B-General_Concept
selection	I-General_Concept
can	O
be	O
extended	O
to	O
the	O
nonparametric	O
case	O
of	O
multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
by	O
choosing	O
a	O
matrix	B-Algorithm
regularization	I-Algorithm
function	O
as	O
this	O
type	O
of	O
norm	O
,	O
it	O
is	O
possible	O
to	O
find	O
a	O
solution	O
that	O
is	O
sparse	O
in	O
terms	O
of	O
which	O
kernels	O
are	O
used	O
,	O
but	O
dense	O
in	O
the	O
coefficient	O
of	O
each	O
used	O
kernel	O
.	O
</s>
<s>
Multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
can	O
also	O
be	O
used	O
as	O
a	O
form	O
of	O
nonlinear	O
variable	B-General_Concept
selection	I-General_Concept
,	O
or	O
as	O
a	O
model	O
aggregation	O
technique	O
(	O
e.g.	O
</s>
