<s>
Sparse	B-Algorithm
coding	O
is	O
a	O
representation	B-General_Concept
learning	I-General_Concept
method	O
which	O
aims	O
at	O
finding	O
a	O
sparse	B-Algorithm
representation	O
of	O
the	O
input	O
data	O
(	O
also	O
known	O
as	O
sparse	B-Algorithm
coding	O
)	O
in	O
the	O
form	O
of	O
a	O
linear	O
combination	O
of	O
basic	O
elements	O
as	O
well	O
as	O
those	O
basic	O
elements	O
themselves	O
.	O
</s>
<s>
Atoms	O
in	O
the	O
dictionary	O
are	O
not	O
required	O
to	O
be	O
orthogonal	B-Algorithm
,	O
and	O
they	O
may	O
be	O
an	O
over-complete	O
spanning	O
set	O
.	O
</s>
<s>
The	O
above	O
two	O
properties	O
lead	O
to	O
having	O
seemingly	O
redundant	O
atoms	O
that	O
allow	O
multiple	O
representations	O
of	O
the	O
same	O
signal	O
but	O
also	O
provide	O
an	O
improvement	O
in	O
sparsity	B-Algorithm
and	O
flexibility	O
of	O
the	O
representation	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
important	O
applications	O
of	O
sparse	B-General_Concept
dictionary	I-General_Concept
learning	I-General_Concept
is	O
in	O
the	O
field	O
of	O
compressed	O
sensing	O
or	O
signal	O
recovery	O
.	O
</s>
<s>
In	O
compressed	O
sensing	O
,	O
a	O
high-dimensional	O
signal	O
can	O
be	O
recovered	O
with	O
only	O
a	O
few	O
linear	O
measurements	O
provided	O
that	O
the	O
signal	O
is	O
sparse	B-Algorithm
or	O
nearly	O
sparse	B-Algorithm
.	O
</s>
<s>
Since	O
not	O
all	O
signals	O
satisfy	O
this	O
sparsity	B-Algorithm
condition	O
,	O
it	O
is	O
of	O
great	O
importance	O
to	O
find	O
a	O
sparse	B-Algorithm
representation	O
of	O
that	O
signal	O
such	O
as	O
the	O
wavelet	B-Algorithm
transform	I-Algorithm
or	O
the	O
directional	O
gradient	O
of	O
a	O
rasterized	O
matrix	O
.	O
</s>
<s>
Once	O
a	O
matrix	O
or	O
a	O
high	O
dimensional	O
vector	O
is	O
transferred	O
to	O
a	O
sparse	B-Algorithm
space	O
,	O
different	O
recovery	O
algorithms	O
like	O
basis	O
pursuit	O
,	O
CoSaMP	O
or	O
fast	O
non-iterative	O
algorithms	O
can	O
be	O
used	O
to	O
recover	O
the	O
signal	O
.	O
</s>
<s>
One	O
of	O
the	O
key	O
principles	O
of	O
dictionary	B-General_Concept
learning	I-General_Concept
is	O
that	O
the	O
dictionary	O
has	O
to	O
be	O
inferred	O
from	O
the	O
input	O
data	O
.	O
</s>
<s>
The	O
emergence	O
of	O
sparse	B-General_Concept
dictionary	I-General_Concept
learning	I-General_Concept
methods	O
was	O
stimulated	O
by	O
the	O
fact	O
that	O
in	O
signal	O
processing	O
one	O
typically	O
wants	O
to	O
represent	O
the	O
input	O
data	O
using	O
as	O
few	O
components	O
as	O
possible	O
.	O
</s>
<s>
Before	O
this	O
approach	O
the	O
general	O
practice	O
was	O
to	O
use	O
predefined	O
dictionaries	O
(	O
such	O
as	O
Fourier	B-Algorithm
or	O
wavelet	B-Algorithm
transforms	I-Algorithm
)	O
.	O
</s>
<s>
However	O
,	O
in	O
certain	O
cases	O
a	O
dictionary	O
that	O
is	O
trained	O
to	O
fit	O
the	O
input	O
data	O
can	O
significantly	O
improve	O
the	O
sparsity	B-Algorithm
,	O
which	O
has	O
applications	O
in	O
data	O
decomposition	O
,	O
compression	O
and	O
analysis	O
and	O
has	O
been	O
used	O
in	O
the	O
fields	O
of	O
image	O
denoising	O
and	O
classification	O
,	O
video	O
and	O
audio	B-Algorithm
processing	I-Algorithm
.	O
</s>
<s>
Sparsity	B-Algorithm
and	O
overcomplete	O
dictionaries	O
have	O
immense	O
applications	O
in	O
image	O
compression	O
,	O
image	O
fusion	O
and	O
inpainting	O
.	O
</s>
<s>
Given	O
the	O
input	O
dataset	O
we	O
wish	O
to	O
find	O
a	O
dictionary	O
and	O
a	O
representation	O
such	O
that	O
both	O
is	O
minimized	O
and	O
the	O
representations	O
are	O
sparse	B-Algorithm
enough	O
.	O
</s>
<s>
controls	O
the	O
trade	O
off	O
between	O
the	O
sparsity	B-Algorithm
and	O
the	O
minimization	O
error	O
.	O
</s>
<s>
In	O
some	O
cases	O
L1-norm	O
is	O
known	O
to	O
ensure	O
sparsity	B-Algorithm
and	O
so	O
the	O
above	O
becomes	O
a	O
convex	O
optimization	O
problem	O
with	O
respect	O
to	O
each	O
of	O
the	O
variables	O
and	O
when	O
the	O
other	O
one	O
is	O
fixed	O
,	O
but	O
it	O
is	O
not	O
jointly	O
convex	O
in	O
.	O
</s>
<s>
The	O
dictionary	O
defined	O
above	O
can	O
be	O
"	O
undercomplete	O
"	O
if	O
or	O
"	O
overcomplete	O
"	O
in	O
case	O
with	O
the	O
latter	O
being	O
a	O
typical	O
assumption	O
for	O
a	O
sparse	B-General_Concept
dictionary	I-General_Concept
learning	I-General_Concept
problem	O
.	O
</s>
<s>
This	O
case	O
is	O
strongly	O
related	O
to	O
dimensionality	B-Algorithm
reduction	I-Algorithm
and	O
techniques	O
like	O
principal	B-Application
component	I-Application
analysis	I-Application
which	O
require	O
atoms	O
to	O
be	O
orthogonal	B-Algorithm
.	O
</s>
<s>
The	O
choice	O
of	O
these	O
subspaces	O
is	O
crucial	O
for	O
efficient	O
dimensionality	B-Algorithm
reduction	I-Algorithm
,	O
but	O
it	O
is	O
not	O
trivial	O
.	O
</s>
<s>
And	O
dimensionality	B-Algorithm
reduction	I-Algorithm
based	O
on	O
dictionary	O
representation	O
can	O
be	O
extended	O
to	O
address	O
specific	O
tasks	O
such	O
as	O
data	O
analysis	O
or	O
classification	O
.	O
</s>
<s>
Overcomplete	O
dictionaries	O
,	O
however	O
,	O
do	O
not	O
require	O
the	O
atoms	O
to	O
be	O
orthogonal	B-Algorithm
(	O
they	O
will	O
never	O
be	O
a	O
basis	O
anyway	O
)	O
thus	O
allowing	O
for	O
more	O
flexible	O
dictionaries	O
and	O
richer	O
data	O
representations	O
.	O
</s>
<s>
An	O
overcomplete	O
dictionary	O
which	O
allows	O
for	O
sparse	B-Algorithm
representation	O
of	O
signal	O
can	O
be	O
a	O
famous	O
transform	O
matrix	O
(	O
wavelets	O
transform	O
,	O
fourier	B-Algorithm
transform	I-Algorithm
)	O
or	O
it	O
can	O
be	O
formulated	O
so	O
that	O
its	O
elements	O
are	O
changed	O
in	O
such	O
a	O
way	O
that	O
it	O
sparsely	O
represents	O
the	O
given	O
signal	O
in	O
a	O
best	O
way	O
.	O
</s>
<s>
As	O
the	O
optimization	O
problem	O
described	O
above	O
can	O
be	O
solved	O
as	O
a	O
convex	O
problem	O
with	O
respect	O
to	O
either	O
dictionary	O
or	O
sparse	B-Algorithm
coding	O
while	O
the	O
other	O
one	O
of	O
the	O
two	O
is	O
fixed	O
,	O
most	O
of	O
the	O
algorithms	O
are	O
based	O
on	O
the	O
idea	O
of	O
iteratively	O
updating	O
one	O
and	O
then	O
the	O
other	O
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
an	O
optimal	O
sparse	B-Algorithm
coding	O
with	O
a	O
given	O
dictionary	O
is	O
known	O
as	O
sparse	B-Algorithm
approximation	O
(	O
or	O
sometimes	O
just	O
sparse	B-Algorithm
coding	O
problem	O
)	O
.	O
</s>
<s>
A	O
number	O
of	O
algorithms	O
have	O
been	O
developed	O
to	O
solve	O
it	O
(	O
such	O
as	O
matching	B-General_Concept
pursuit	I-General_Concept
and	O
LASSO	B-Algorithm
)	O
and	O
are	O
incorporated	O
in	O
the	O
algorithms	O
described	O
below	O
.	O
</s>
<s>
The	O
method	O
of	O
optimal	O
directions	O
(	O
or	O
MOD	O
)	O
was	O
one	O
of	O
the	O
first	O
methods	O
introduced	O
to	O
tackle	O
the	O
sparse	B-General_Concept
dictionary	I-General_Concept
learning	I-General_Concept
problem	O
.	O
</s>
<s>
MOD	O
alternates	O
between	O
getting	O
the	O
sparse	B-Algorithm
coding	O
using	O
a	O
method	O
such	O
as	O
matching	B-General_Concept
pursuit	I-General_Concept
and	O
updating	O
the	O
dictionary	O
by	O
computing	O
the	O
analytical	O
solution	O
of	O
the	O
problem	O
given	O
by	O
where	O
is	O
a	O
Moore-Penrose	O
pseudoinverse	O
.	O
</s>
<s>
After	O
this	O
update	O
is	O
renormalized	O
to	O
fit	O
the	O
constraints	O
and	O
the	O
new	O
sparse	B-Algorithm
coding	O
is	O
obtained	O
again	O
.	O
</s>
<s>
This	O
shortcoming	O
has	O
inspired	O
the	O
development	O
of	O
other	O
dictionary	B-General_Concept
learning	I-General_Concept
methods	O
.	O
</s>
<s>
K-SVD	O
is	O
an	O
algorithm	O
that	O
performs	O
SVD	O
at	O
its	O
core	O
to	O
update	O
the	O
atoms	O
of	O
the	O
dictionary	O
one	O
by	O
one	O
and	O
basically	O
is	O
a	O
generalization	O
of	O
K-means	B-Algorithm
.	O
</s>
<s>
This	O
algorithm	O
's	O
essence	O
is	O
to	O
first	O
fix	O
the	O
dictionary	O
,	O
find	O
the	O
best	O
possible	O
under	O
the	O
above	O
constraint	O
(	O
using	O
Orthogonal	B-Algorithm
Matching	B-General_Concept
Pursuit	I-General_Concept
)	O
and	O
then	O
iteratively	O
update	O
the	O
atoms	O
of	O
dictionary	O
in	O
the	O
following	O
manner	O
:	O
</s>
<s>
The	O
next	O
steps	O
of	O
the	O
algorithm	O
include	O
rank-1	O
approximation	O
of	O
the	O
residual	O
matrix	O
,	O
updating	O
and	O
enforcing	O
the	O
sparsity	B-Algorithm
of	O
after	O
the	O
update	O
.	O
</s>
<s>
This	O
algorithm	O
is	O
considered	O
to	O
be	O
standard	O
for	O
dictionary	B-General_Concept
learning	I-General_Concept
and	O
is	O
used	O
in	O
a	O
variety	O
of	O
applications	O
.	O
</s>
<s>
An	O
algorithm	O
based	O
on	O
solving	O
a	O
dual	B-Algorithm
Lagrangian	I-Algorithm
problem	I-Algorithm
provides	O
an	O
efficient	O
way	O
to	O
solve	O
for	O
the	O
dictionary	O
having	O
no	O
complications	O
induced	O
by	O
the	O
sparsity	B-Algorithm
function	O
.	O
</s>
<s>
After	O
applying	O
one	O
of	O
the	O
optimization	O
methods	O
to	O
the	O
value	O
of	O
the	O
dual	O
(	O
such	O
as	O
Newton	B-Algorithm
's	I-Algorithm
method	I-Algorithm
or	O
conjugate	B-Algorithm
gradient	I-Algorithm
)	O
we	O
get	O
the	O
value	O
of	O
:	O
</s>
<s>
Solving	O
this	O
problem	O
is	O
less	O
computational	O
hard	O
because	O
the	O
amount	O
of	O
dual	O
variables	O
is	O
a	O
lot	O
of	O
times	O
much	O
less	O
than	O
the	O
amount	O
of	O
variables	O
in	O
the	O
primal	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
,	O
where	O
is	O
the	O
permitted	O
error	O
in	O
the	O
reconstruction	O
LASSO	B-Algorithm
.	O
</s>
<s>
,	O
where	O
controls	O
the	O
trade-off	O
between	O
sparsity	B-Algorithm
and	O
the	O
reconstruction	O
error	O
.	O
</s>
<s>
This	O
method	O
focuses	O
on	O
constructing	O
a	O
dictionary	O
that	O
is	O
composed	O
of	O
differently	O
scaled	O
dictionaries	O
to	O
improve	O
sparsity	B-Algorithm
.	O
</s>
<s>
Sparse	B-Algorithm
dictionaries	O
.	O
</s>
<s>
This	O
method	O
focuses	O
on	O
not	O
only	O
providing	O
a	O
sparse	B-Algorithm
representation	O
but	O
also	O
constructing	O
a	O
sparse	B-Algorithm
dictionary	O
which	O
is	O
enforced	O
by	O
the	O
expression	O
where	O
is	O
some	O
pre-defined	O
analytical	O
dictionary	O
with	O
desirable	O
properties	O
such	O
as	O
fast	O
computation	O
and	O
is	O
a	O
sparse	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Such	O
formulation	O
allows	O
to	O
directly	O
combine	O
the	O
fast	O
implementation	O
of	O
analytical	O
dictionaries	O
with	O
the	O
flexibility	O
of	O
sparse	B-Algorithm
approaches	O
.	O
</s>
<s>
Many	O
common	O
approaches	O
to	O
sparse	B-General_Concept
dictionary	I-General_Concept
learning	I-General_Concept
rely	O
on	O
the	O
fact	O
that	O
the	O
whole	O
input	O
data	O
(	O
or	O
at	O
least	O
a	O
large	O
enough	O
training	O
dataset	O
)	O
is	O
available	O
for	O
the	O
algorithm	O
.	O
</s>
<s>
The	O
other	O
case	O
where	O
this	O
assumption	O
can	O
not	O
be	O
made	O
is	O
when	O
the	O
input	O
data	O
comes	O
in	O
a	O
form	O
of	O
a	O
stream	B-Architecture
.	O
</s>
<s>
Such	O
cases	O
lie	O
in	O
the	O
field	O
of	O
study	O
of	O
online	B-Algorithm
learning	I-Algorithm
which	O
essentially	O
suggests	O
iteratively	O
updating	O
the	O
model	O
upon	O
the	O
new	O
data	O
points	O
becoming	O
available	O
.	O
</s>
<s>
Find	O
a	O
sparse	B-Algorithm
coding	O
using	O
LARS	O
:	O
</s>
<s>
Update	O
dictionary	O
using	O
block-coordinate	B-Algorithm
approach	O
:	O
</s>
<s>
This	O
method	O
allows	O
us	O
to	O
gradually	O
update	O
the	O
dictionary	O
as	O
new	O
data	O
becomes	O
available	O
for	O
sparse	B-Algorithm
representation	B-General_Concept
learning	I-General_Concept
and	O
helps	O
drastically	O
reduce	O
the	O
amount	O
of	O
memory	O
needed	O
to	O
store	O
the	O
dataset	O
(	O
which	O
often	O
has	O
a	O
huge	O
size	O
)	O
.	O
</s>
<s>
The	O
dictionary	B-General_Concept
learning	I-General_Concept
framework	O
,	O
namely	O
the	O
linear	O
decomposition	O
of	O
an	O
input	O
signal	O
using	O
a	O
few	O
basis	O
elements	O
learned	O
from	O
data	O
itself	O
,	O
has	O
led	O
to	O
state-of-art	O
results	O
in	O
various	O
image	O
and	O
video	O
processing	O
tasks	O
.	O
</s>
<s>
It	O
also	O
has	O
properties	O
that	O
are	O
useful	O
for	O
signal	O
denoising	O
since	O
usually	O
one	O
can	O
learn	O
a	O
dictionary	O
to	O
represent	O
the	O
meaningful	O
part	O
of	O
the	O
input	O
signal	O
in	O
a	O
sparse	B-Algorithm
way	O
but	O
the	O
noise	O
in	O
the	O
input	O
will	O
have	O
a	O
much	O
less	O
sparse	B-Algorithm
representation	O
.	O
</s>
<s>
Sparse	B-General_Concept
dictionary	I-General_Concept
learning	I-General_Concept
has	O
been	O
successfully	O
applied	O
to	O
various	O
image	O
,	O
video	O
and	O
audio	B-Algorithm
processing	I-Algorithm
tasks	O
as	O
well	O
as	O
to	O
texture	O
synthesis	O
and	O
unsupervised	O
clustering	O
.	O
</s>
<s>
In	O
evaluations	O
with	O
the	O
Bag-of-Words	B-General_Concept
model	O
,	O
sparse	B-Algorithm
coding	O
was	O
found	O
empirically	O
to	O
outperform	O
other	O
coding	O
approaches	O
on	O
the	O
object	O
category	O
recognition	O
tasks	O
.	O
</s>
<s>
Dictionary	B-General_Concept
learning	I-General_Concept
is	O
used	O
to	O
analyse	O
medical	O
signals	O
in	O
detail	O
.	O
</s>
