<s>
Persistent	B-Algorithm
homology	I-Algorithm
is	O
a	O
method	O
for	O
computing	O
topological	O
features	O
of	O
a	O
space	O
at	O
different	O
spatial	O
resolutions	O
.	O
</s>
<s>
To	O
find	O
the	O
persistent	B-Algorithm
homology	I-Algorithm
of	O
a	O
space	O
,	O
the	O
space	O
must	O
first	O
be	O
represented	O
as	O
a	O
simplicial	O
complex	O
.	O
</s>
<s>
One	O
common	O
method	O
of	O
doing	O
this	O
is	O
via	O
taking	O
the	O
sublevel	O
filtration	O
of	O
the	O
distance	O
to	O
a	O
point	B-Algorithm
cloud	I-Algorithm
,	O
or	O
equivalently	O
,	O
the	O
offset	O
filtration	O
on	O
the	O
point	B-Algorithm
cloud	I-Algorithm
and	O
taking	O
its	O
nerve	O
in	O
order	O
to	O
get	O
the	O
simplicial	O
filtration	O
known	O
as	O
Čech	O
filtration	O
.	O
</s>
<s>
When	O
,	O
the	O
inclusion	O
induces	O
a	O
homomorphism	O
on	O
the	O
simplicial	B-Algorithm
homology	I-Algorithm
groups	O
for	O
each	O
dimension	O
.	O
</s>
<s>
The	O
persistent	B-Algorithm
homology	I-Algorithm
groups	O
are	O
the	O
images	O
of	O
these	O
homomorphisms	O
,	O
and	O
the	O
persistent	O
Betti	O
numbers	O
are	O
the	O
ranks	O
of	O
those	O
groups	O
.	O
</s>
<s>
the	O
size	O
function	O
,	O
a	O
predecessor	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
.	O
</s>
<s>
Equivalently	O
,	O
we	O
may	O
consider	O
it	O
as	O
a	O
functor	B-Language
from	O
considered	O
as	O
a	O
category	O
to	O
the	O
category	O
of	O
vector	O
spaces	O
(	O
or	O
-modules	O
)	O
.	O
</s>
<s>
Each	O
of	O
these	O
two	O
theorems	O
allows	O
us	O
to	O
uniquely	O
represent	O
the	O
persistent	B-Algorithm
homology	I-Algorithm
of	O
a	O
filtered	O
simplicial	O
complex	O
with	O
a	O
persistence	O
barcode	O
or	O
persistence	O
diagram	O
.	O
</s>
<s>
Persistent	B-Algorithm
homology	I-Algorithm
is	O
stable	O
in	O
a	O
precise	O
sense	O
,	O
which	O
provides	O
robustness	O
against	O
noise	O
.	O
</s>
