<s>
The	O
method	O
of	O
iteratively	B-Algorithm
reweighted	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
(	O
IRLS	B-Algorithm
)	O
is	O
used	O
to	O
solve	O
certain	O
optimization	O
problems	O
with	O
objective	O
functions	O
of	O
the	O
form	O
of	O
a	O
p-norm	O
:	O
</s>
<s>
by	O
an	O
iterative	B-Algorithm
method	I-Algorithm
in	O
which	O
each	O
step	O
involves	O
solving	O
a	O
weighted	B-Algorithm
least	I-Algorithm
squares	I-Algorithm
problem	O
of	O
the	O
form	O
:	O
</s>
<s>
IRLS	B-Algorithm
is	O
used	O
to	O
find	O
the	O
maximum	O
likelihood	O
estimates	O
of	O
a	O
generalized	O
linear	O
model	O
,	O
and	O
in	O
robust	O
regression	O
to	O
find	O
an	O
M-estimator	O
,	O
as	O
a	O
way	O
of	O
mitigating	O
the	O
influence	O
of	O
outliers	O
in	O
an	O
otherwise	O
normally-distributed	O
data	O
set	O
,	O
for	O
example	O
,	O
by	O
minimizing	O
the	O
least	B-General_Concept
absolute	I-General_Concept
errors	I-General_Concept
rather	O
than	O
the	O
least	B-Algorithm
square	I-Algorithm
errors	I-Algorithm
.	O
</s>
<s>
One	O
of	O
the	O
advantages	O
of	O
IRLS	B-Algorithm
over	O
linear	B-Algorithm
programming	I-Algorithm
and	O
convex	O
programming	O
is	O
that	O
it	O
can	O
be	O
used	O
with	O
Gauss	B-Algorithm
–	I-Algorithm
Newton	I-Algorithm
and	O
Levenberg	B-Algorithm
–	I-Algorithm
Marquardt	I-Algorithm
numerical	O
algorithms	O
.	O
</s>
<s>
IRLS	B-Algorithm
can	O
be	O
used	O
for	O
ℓ1	O
minimization	O
and	O
smoothed	O
ℓp	O
minimization	O
,	O
p	O
<	O
1	O
,	O
in	O
compressed	O
sensing	O
problems	O
.	O
</s>
<s>
It	O
has	O
been	O
proved	O
that	O
the	O
algorithm	O
has	O
a	O
linear	O
rate	O
of	O
convergence	O
for	O
ℓ1	O
norm	O
and	O
superlinear	O
for	O
ℓt	O
with	O
t	O
<	O
1	O
,	O
under	O
the	O
restricted	B-Algorithm
isometry	I-Algorithm
property	I-Algorithm
,	O
which	O
is	O
generally	O
a	O
sufficient	O
condition	O
for	O
sparse	O
solutions	O
.	O
</s>
<s>
However	O
,	O
in	O
most	O
practical	O
situations	O
,	O
the	O
restricted	B-Algorithm
isometry	I-Algorithm
property	I-Algorithm
is	O
not	O
satisfied	O
.	O
</s>
<s>
To	O
find	O
the	O
parameters	O
β	O
=(	O
β1	O
,	O
…,	O
βk	O
)	O
T	O
which	O
minimize	O
the	O
Lp	O
norm	O
for	O
the	O
linear	B-General_Concept
regression	I-General_Concept
problem	O
,	O
</s>
<s>
the	O
IRLS	B-Algorithm
algorithm	O
at	O
step	O
t+1	O
involves	O
solving	O
the	O
weighted	B-Algorithm
linear	I-Algorithm
least	I-Algorithm
squares	I-Algorithm
problem	O
:	O
</s>
<s>
where	O
W(t )	O
is	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
weights	O
,	O
usually	O
with	O
all	O
elements	O
set	O
initially	O
to	O
:	O
</s>
<s>
In	O
the	O
case	O
p	O
=	O
1	O
,	O
this	O
corresponds	O
to	O
least	B-General_Concept
absolute	I-General_Concept
deviation	I-General_Concept
regression	O
(	O
in	O
this	O
case	O
,	O
the	O
problem	O
would	O
be	O
better	O
approached	O
by	O
use	O
of	O
linear	B-Algorithm
programming	I-Algorithm
methods	O
,	O
so	O
the	O
result	O
would	O
be	O
exact	O
)	O
and	O
the	O
formula	O
is	O
:	O
</s>
