<s>
In	O
mathematics	O
,	O
more	O
specifically	O
in	O
linear	B-Language
algebra	I-Language
,	O
the	O
spark	B-Algorithm
of	O
a	O
matrix	B-Architecture
is	O
the	O
smallest	O
integer	O
such	O
that	O
there	O
exists	O
a	O
set	O
of	O
columns	O
in	O
which	O
are	O
linearly	O
dependent	O
.	O
</s>
<s>
The	O
concept	O
of	O
matrix	B-Architecture
spark	B-Algorithm
finds	O
applications	O
in	O
error-correction	B-Error_Name
codes	I-Error_Name
,	O
compressive	O
sensing	O
,	O
and	O
matroid	O
theory	O
,	O
and	O
provides	O
a	O
simple	O
criterion	O
for	O
maximal	O
sparsity	B-Algorithm
of	O
solutions	O
to	O
a	O
system	O
of	O
linear	O
equations	O
.	O
</s>
<s>
The	O
spark	B-Algorithm
of	O
a	O
matrix	B-Architecture
is	O
NP-hard	O
to	O
compute	O
.	O
</s>
<s>
Formally	O
,	O
the	O
spark	B-Algorithm
of	O
a	O
matrix	B-Architecture
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
Equivalently	O
,	O
the	O
spark	B-Algorithm
of	O
a	O
matrix	B-Architecture
is	O
the	O
size	O
of	O
its	O
smallest	O
circuit	O
(	O
a	O
subset	O
of	O
column	O
indices	O
such	O
that	O
has	O
a	O
nonzero	O
solution	O
,	O
but	O
every	O
subset	O
of	O
it	O
does	O
not	O
)	O
.	O
</s>
<s>
By	O
contrast	O
,	O
the	O
rank	O
of	O
a	O
matrix	B-Architecture
is	O
the	O
largest	O
number	O
such	O
that	O
some	O
set	O
of	O
columns	O
of	O
is	O
linearly	O
independent	O
.	O
</s>
<s>
Consider	O
the	O
following	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
spark	B-Algorithm
of	O
this	O
matrix	B-Architecture
equals	O
3	O
because	O
:	O
</s>
<s>
If	O
,	O
the	O
following	O
simple	O
properties	O
hold	O
for	O
the	O
spark	B-Algorithm
of	O
a	O
matrix	B-Architecture
:	O
</s>
<s>
(	O
If	O
the	O
spark	B-Algorithm
equals	O
,	O
then	O
the	O
matrix	B-Architecture
has	O
full	O
rank	O
.	O
)	O
</s>
<s>
if	O
and	O
only	O
if	O
the	O
matrix	B-Architecture
has	O
a	O
zero	O
column	O
.	O
</s>
<s>
The	O
spark	B-Algorithm
yields	O
a	O
simple	O
criterion	O
for	O
uniqueness	O
of	O
sparse	O
solutions	O
of	O
linear	O
equation	O
systems	O
.	O
</s>
<s>
If	O
this	O
system	O
has	O
a	O
solution	O
that	O
satisfies	O
,	O
then	O
this	O
solution	O
is	O
the	O
sparsest	B-Algorithm
possible	O
solution	O
.	O
</s>
<s>
If	O
the	O
columns	O
of	O
the	O
matrix	B-Architecture
are	O
normalized	O
to	O
unit	O
norm	O
,	O
we	O
can	O
lower	O
bound	O
its	O
spark	B-Algorithm
in	O
terms	O
of	O
its	O
dictionary	O
coherence	O
:	O
</s>
<s>
The	O
minimum	O
distance	O
of	O
a	O
linear	O
code	O
equals	O
the	O
spark	B-Algorithm
of	O
its	O
parity-check	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
concept	O
of	O
the	O
spark	B-Algorithm
is	O
also	O
of	O
use	O
in	O
the	O
theory	O
of	O
compressive	O
sensing	O
,	O
where	O
requirements	O
on	O
the	O
spark	B-Algorithm
of	O
the	O
measurement	O
matrix	B-Architecture
are	O
used	O
to	O
ensure	O
stability	O
and	O
consistency	O
of	O
various	O
estimation	O
techniques	O
.	O
</s>
<s>
It	O
is	O
also	O
known	O
in	O
matroid	O
theory	O
as	O
the	O
girth	O
of	O
the	O
vector	O
matroid	O
associated	O
with	O
the	O
columns	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
The	O
spark	B-Algorithm
of	O
a	O
matrix	B-Architecture
is	O
NP-hard	O
to	O
compute	O
.	O
</s>
