<s>
Structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
is	O
a	O
class	O
of	O
methods	O
,	O
and	O
an	O
area	O
of	O
research	O
in	O
statistical	B-General_Concept
learning	I-General_Concept
theory	I-General_Concept
,	O
that	O
extend	O
and	O
generalize	O
sparsity	O
regularization	O
learning	O
methods	O
.	O
</s>
<s>
Both	O
sparsity	O
and	O
structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
methods	O
seek	O
to	O
exploit	O
the	O
assumption	O
that	O
the	O
output	O
variable	O
(	O
i.e.	O
,	O
response	O
,	O
or	O
dependent	O
variable	O
)	O
to	O
be	O
learned	O
can	O
be	O
described	O
by	O
a	O
reduced	O
number	O
of	O
variables	O
in	O
the	O
input	O
space	O
(	O
i.e.	O
,	O
the	O
domain	B-Algorithm
,	O
space	O
of	O
features	B-Algorithm
or	O
explanatory	O
variables	O
)	O
.	O
</s>
<s>
Structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
methods	O
generalize	O
and	O
extend	O
sparsity	O
regularization	O
methods	O
,	O
by	O
allowing	O
for	O
optimal	O
selection	O
over	O
structures	O
like	O
groups	O
or	O
networks	O
of	O
input	O
variables	O
in	O
.	O
</s>
<s>
Common	O
motivation	O
for	O
the	O
use	O
of	O
structured	O
sparsity	O
methods	O
are	O
model	O
interpretability	O
,	O
high-dimensional	B-Algorithm
learning	I-Algorithm
(	O
where	O
dimensionality	O
of	O
may	O
be	O
higher	O
than	O
the	O
number	O
of	O
observations	O
)	O
,	O
and	O
reduction	O
of	O
computational	O
complexity	O
.	O
</s>
<s>
Consider	O
the	O
linear	O
kernel	O
regularized	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
problem	O
with	O
a	O
loss	O
function	O
and	O
the	O
"	O
norm	O
"	O
as	O
the	O
regularization	O
penalty	O
:	O
</s>
<s>
Structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
extends	O
and	O
generalizes	O
the	O
variable	B-General_Concept
selection	I-General_Concept
problem	O
that	O
characterizes	O
sparsity	O
regularization	O
.	O
</s>
<s>
Consider	O
the	O
above	O
regularized	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
problem	O
with	O
a	O
general	O
kernel	O
and	O
associated	O
feature	O
map	O
with	O
.	O
</s>
<s>
Structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
methods	O
allow	O
to	O
impose	O
such	O
structure	O
by	O
adding	O
structure	O
to	O
the	O
norms	O
defining	O
the	O
regularization	O
term	O
.	O
</s>
<s>
The	O
above	O
norm	O
is	O
also	O
referred	O
to	O
as	O
group	O
Lasso	B-Algorithm
.	O
</s>
<s>
Consider	O
again	O
the	O
group	O
Lasso	B-Algorithm
for	O
a	O
regularized	O
empirical	B-General_Concept
risk	I-General_Concept
minimization	I-General_Concept
problem	O
:	O
</s>
<s>
As	O
in	O
the	O
non-overlapping	O
groups	O
case	O
,	O
the	O
group	O
Lasso	B-Algorithm
regularizer	O
will	O
potentially	O
set	O
entire	O
groups	O
of	O
coefficients	O
to	O
zero	O
.	O
</s>
<s>
A	O
different	O
approach	O
is	O
to	O
consider	O
union	O
of	O
groups	O
for	O
variable	B-General_Concept
selection	I-General_Concept
.	O
</s>
<s>
The	O
objective	O
function	O
using	O
group	O
lasso	B-Algorithm
consists	O
of	O
an	O
error	O
function	O
,	O
which	O
is	O
generally	O
required	O
to	O
be	O
convex	O
but	O
not	O
necessarily	O
strongly	O
convex	O
,	O
and	O
a	O
group	O
regularization	O
term	O
.	O
</s>
<s>
An	O
example	O
of	O
a	O
way	O
to	O
fix	O
this	O
is	O
to	O
introduce	O
the	O
squared	O
norm	O
of	O
the	O
weight	O
vector	O
as	O
an	O
additional	O
regularization	O
term	O
while	O
keeping	O
the	O
regularization	O
term	O
from	O
the	O
group	O
lasso	B-Algorithm
approach	O
.	O
</s>
<s>
Provided	O
that	O
the	O
coefficient	O
is	O
suitably	O
small	O
but	O
still	O
positive	O
,	O
the	O
weight	O
vector	O
minimizing	O
the	O
resulting	O
objective	O
function	O
is	O
generally	O
very	O
close	O
to	O
a	O
weight	O
vector	O
that	O
minimizes	O
the	O
objective	O
function	O
that	O
would	O
result	O
from	O
removing	O
the	O
group	O
regularization	O
term	O
altogether	O
from	O
the	O
original	O
objective	O
function	O
;	O
the	O
latter	O
scenario	O
corresponds	O
to	O
the	O
group	O
Lasso	B-Algorithm
approach	O
.	O
</s>
<s>
These	O
norms	O
arise	O
from	O
submodular	B-Algorithm
functions	I-Algorithm
and	O
allow	O
the	O
incorporation	O
of	O
prior	O
assumptions	O
on	O
the	O
structure	O
of	O
the	O
input	O
variables	O
.	O
</s>
<s>
Unsupervised	B-General_Concept
learning	I-General_Concept
methods	O
are	O
often	O
used	O
to	O
learn	O
the	O
parameters	O
of	O
latent	O
variable	O
models	O
.	O
</s>
<s>
If	O
the	O
structure	O
assumed	O
over	O
variables	O
is	O
in	O
the	O
form	O
of	O
a	O
1D	O
,	O
2D	O
or	O
3D	O
grid	O
,	O
then	O
submodular	B-Algorithm
functions	I-Algorithm
based	O
on	O
overlapping	O
groups	O
can	O
be	O
considered	O
as	O
norms	O
,	O
leading	O
to	O
stable	O
sets	O
equal	O
to	O
rectangular	O
or	O
convex	O
shapes	O
.	O
</s>
<s>
Two	O
main	O
approaches	O
for	O
solving	O
the	O
optimization	O
problem	O
are	O
:	O
1	O
)	O
greedy	O
methods	O
,	O
such	O
as	O
step-wise	O
regression	O
in	O
statistics	O
,	O
or	O
matching	B-General_Concept
pursuit	I-General_Concept
in	O
signal	O
processing	O
;	O
and	O
2	O
)	O
convex	O
relaxation	O
formulation	O
approaches	O
and	O
proximal	B-Algorithm
gradient	I-Algorithm
optimization	O
methods	O
.	O
</s>
<s>
Such	O
a	O
scheme	O
is	O
called	O
basis	O
pursuit	O
or	O
the	O
Lasso	B-Algorithm
,	O
which	O
substitutes	O
the	O
"	O
norm	O
"	O
for	O
the	O
convex	O
,	O
non-differentiable	O
norm	O
.	O
</s>
<s>
Proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
,	O
also	O
called	O
forward-backward	O
splitting	O
,	O
are	O
optimization	O
methods	O
useful	O
for	O
minimizing	O
functions	O
with	O
a	O
convex	O
and	O
differentiable	O
component	O
,	O
and	O
a	O
convex	O
potentially	O
non-differentiable	O
component	O
.	O
</s>
<s>
As	O
such	O
,	O
proximal	B-Algorithm
gradient	I-Algorithm
methods	I-Algorithm
are	O
useful	O
for	O
solving	O
sparsity	O
and	O
structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
problems	O
of	O
the	O
following	O
form	O
:	O
</s>
<s>
Structured	B-Algorithm
Sparsity	I-Algorithm
regularization	I-Algorithm
can	O
be	O
applied	O
in	O
the	O
context	O
of	O
multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
Multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
refers	O
to	O
a	O
set	O
of	O
machine	O
learning	O
methods	O
that	O
use	O
a	O
predefined	O
set	O
of	O
kernels	O
and	O
learn	O
an	O
optimal	O
linear	O
or	O
non-linear	O
combination	O
of	O
kernels	O
as	O
part	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
The	O
norm	O
mentioned	O
above	O
can	O
be	O
seen	O
as	O
the	O
group	O
norm	O
in	O
associated	O
to	O
the	O
subspaces	O
,	O
,	O
providing	O
a	O
connection	O
to	O
structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
.	O
</s>
<s>
In	O
the	O
structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
approach	O
to	O
this	O
scenario	O
,	O
the	O
relevant	O
groups	O
of	O
variables	O
which	O
the	O
group	O
norms	O
consider	O
correspond	O
to	O
the	O
subspaces	O
and	O
.	O
</s>
<s>
This	O
approach	O
promotes	O
setting	O
the	O
groups	O
of	O
coefficients	O
corresponding	O
to	O
these	O
subspaces	O
to	O
zero	O
as	O
opposed	O
to	O
only	O
individual	O
coefficients	O
,	O
promoting	O
sparse	O
multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
.	O
</s>
<s>
Considering	O
sparse	O
multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
is	O
useful	O
in	O
several	O
situations	O
including	O
the	O
following	O
:	O
</s>
<s>
Nonlinear	O
variable	B-General_Concept
selection	I-General_Concept
:	O
Consider	O
kernels	O
depending	O
only	O
one	O
dimension	O
of	O
the	O
input	O
.	O
</s>
<s>
Generally	O
sparse	O
multiple	B-Algorithm
kernel	I-Algorithm
learning	I-Algorithm
is	O
particularly	O
useful	O
when	O
there	O
are	O
many	O
kernels	O
and	O
model	O
selection	O
and	O
interpretability	O
are	O
important	O
.	O
</s>
<s>
Structured	B-Algorithm
sparsity	I-Algorithm
regularization	I-Algorithm
methods	O
have	O
been	O
used	O
in	O
a	O
number	O
of	O
settings	O
where	O
it	O
is	O
desired	O
to	O
impose	O
an	O
a	O
priori	O
input	O
variable	O
structure	O
to	O
the	O
regularization	O
process	O
.	O
</s>
