<s>
The	O
Kaczmarz	B-Algorithm
method	I-Algorithm
or	O
Kaczmarz	O
's	O
algorithm	O
is	O
an	O
iterative	B-Algorithm
algorithm	I-Algorithm
for	O
solving	O
linear	O
equation	O
systems	O
.	O
</s>
<s>
It	O
was	O
first	O
discovered	O
by	O
the	O
Polish	O
mathematician	O
Stefan	O
Kaczmarz	O
,	O
and	O
was	O
rediscovered	O
in	O
the	O
field	O
of	O
image	O
reconstruction	O
from	O
projections	O
by	O
Richard	O
Gordon	O
,	O
Robert	O
Bender	O
,	O
and	O
Gabor	O
Herman	O
in	O
1970	O
,	O
where	O
it	O
is	O
called	O
the	O
Algebraic	B-Algorithm
Reconstruction	I-Algorithm
Technique	I-Algorithm
(	O
ART	O
)	O
.	O
</s>
<s>
The	O
Kaczmarz	B-Algorithm
method	I-Algorithm
is	O
applicable	O
to	O
any	O
linear	O
system	O
of	O
equations	O
,	O
but	O
its	O
computational	O
advantage	O
relative	O
to	O
other	O
methods	O
depends	O
on	O
the	O
system	O
being	O
sparse	B-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
system	O
of	O
linear	O
equations	O
,	O
let	O
be	O
the	O
number	O
of	O
rows	O
of	O
A	O
,	O
be	O
the	O
th	O
row	O
of	O
complex-valued	O
matrix	B-Architecture
,	O
and	O
let	O
be	O
arbitrary	O
complex-valued	O
initial	O
approximation	O
to	O
the	O
solution	O
of	O
.	O
</s>
<s>
There	O
are	O
versions	O
of	O
the	O
method	O
that	O
converge	O
to	O
a	O
regularized	O
weighted	O
least	O
squares	O
solution	O
when	O
applied	O
to	O
a	O
system	O
of	O
inconsistent	O
equations	O
and	O
,	O
at	O
least	O
as	O
far	O
as	O
initial	O
behavior	O
is	O
concerned	O
,	O
at	O
a	O
lesser	O
cost	O
than	O
other	O
iterative	B-Algorithm
methods	I-Algorithm
,	O
such	O
as	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
This	O
method	O
can	O
be	O
seen	O
as	O
a	O
particular	O
case	O
of	O
stochastic	B-Algorithm
gradient	I-Algorithm
descent	I-Algorithm
.	O
</s>
<s>
Under	O
such	O
circumstances	O
converges	O
exponentially	O
fast	O
to	O
the	O
solution	O
of	O
and	O
the	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
depends	O
only	O
on	O
the	O
scaled	O
condition	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
In	O
2015	O
,	O
Robert	O
M	O
.	O
Gower	O
and	O
Peter	O
Richtarik	O
developed	O
a	O
versatile	O
randomized	O
iterative	B-Algorithm
method	I-Algorithm
for	O
solving	O
a	O
consistent	O
system	O
of	O
linear	O
equations	O
which	O
includes	O
the	O
randomized	O
Kaczmarz	O
algorithm	O
as	O
a	O
special	O
case	O
.	O
</s>
<s>
The	O
method	O
is	O
shown	O
to	O
enjoy	O
exponential	O
rate	O
decay	O
(	O
in	O
expectation	O
)	O
-	O
also	O
known	O
as	O
linear	B-Architecture
convergence	I-Architecture
,	O
under	O
very	O
mild	O
conditions	O
on	O
the	O
way	O
randomness	O
enters	O
the	O
algorithm	O
.	O
</s>
<s>
Interesting	O
new	O
insights	O
about	O
the	O
randomized	O
Kaczmarz	B-Algorithm
method	I-Algorithm
that	O
can	O
be	O
gained	O
from	O
the	O
analysis	O
of	O
the	O
method	O
include	O
:	O
</s>
<s>
The	O
general	O
rate	O
of	O
the	O
Gower-Richtarik	O
algorithm	O
precisely	O
recovers	O
the	O
rate	O
of	O
the	O
randomized	O
Kaczmarz	B-Algorithm
method	I-Algorithm
in	O
the	O
special	O
case	O
when	O
it	O
reduced	O
to	O
it	O
.	O
</s>
<s>
However	O
,	O
the	O
amount	O
by	O
which	O
it	O
is	O
better	O
depends	O
on	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
When	O
applied	O
to	O
a	O
system	O
with	O
matrix	B-Architecture
which	O
is	O
positive	O
definite	O
,	O
Randomized	O
Kaczmarz	B-Algorithm
method	I-Algorithm
is	O
equivalent	O
to	O
the	O
Stochastic	B-Algorithm
Gradient	I-Algorithm
Descent	I-Algorithm
(	O
SGD	O
)	O
method	O
(	O
with	O
a	O
very	O
special	O
stepsize	O
)	O
for	O
minimizing	O
the	O
strongly	O
convex	O
quadratic	O
function	O
Note	O
that	O
since	O
is	O
convex	O
,	O
the	O
minimizers	O
of	O
must	O
satisfy	O
,	O
which	O
is	O
equivalent	O
to	O
The	O
"	O
special	O
stepsize	O
"	O
is	O
the	O
stepsize	O
which	O
leads	O
to	O
a	O
point	O
which	O
in	O
the	O
one-dimensional	O
line	O
spanned	O
by	O
the	O
stochastic	O
gradient	O
minimizes	O
the	O
Euclidean	O
distance	O
from	O
the	O
unknown( 	O
!	O
)	O
</s>
<s>
and	O
a	O
random	O
matrix	B-Architecture
with	O
as	O
many	O
rows	O
as	O
(	O
and	O
possibly	O
random	O
number	O
of	O
columns	O
)	O
.	O
</s>
<s>
Randomized	O
Kaczmarz	B-Algorithm
method	I-Algorithm
is	O
obtained	O
by	O
picking	O
to	O
be	O
the	O
identity	O
matrix	B-Architecture
,	O
and	O
to	O
be	O
the	O
unit	O
coordinate	O
vector	O
with	O
probability	O
Different	O
choices	O
of	O
and	O
lead	O
to	O
different	O
variants	O
of	O
the	O
method	O
.	O
</s>
<s>
where	O
by	O
we	O
denote	O
the	O
Moore-Penrose	O
pseudo-inverse	O
of	O
matrix	B-Architecture
.	O
</s>
<s>
where	O
is	O
the	O
identity	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
iteration	O
matrix	B-Architecture
,	O
is	O
random	O
,	O
whence	O
the	O
name	O
of	O
this	O
formulation	O
.	O
</s>
<s>
Since	O
the	O
convergence	O
of	O
the	O
(	O
randomized	O
)	O
Kaczmarz	B-Algorithm
method	I-Algorithm
depends	O
on	O
a	O
rate	B-Architecture
of	I-Architecture
convergence	I-Architecture
the	O
method	O
may	O
make	O
slow	O
progress	O
on	O
some	O
practical	O
problems	O
.	O
</s>
