<s>
In	O
applied	O
mathematics	O
,	O
topological	B-Algorithm
data	I-Algorithm
analysis	I-Algorithm
(	O
TDA	O
)	O
is	O
an	O
approach	O
to	O
the	O
analysis	O
of	O
datasets	O
using	O
techniques	O
from	O
topology	B-Architecture
.	O
</s>
<s>
TDA	O
provides	O
a	O
general	O
framework	O
to	O
analyze	O
such	O
data	O
in	O
a	O
manner	O
that	O
is	O
insensitive	O
to	O
the	O
particular	O
metric	O
chosen	O
and	O
provides	O
dimensionality	B-Algorithm
reduction	I-Algorithm
and	O
robustness	O
to	O
noise	O
.	O
</s>
<s>
TDA	O
has	O
combined	O
algebraic	O
topology	B-Architecture
and	O
other	O
tools	O
from	O
pure	O
mathematics	O
to	O
allow	O
mathematically	O
rigorous	O
study	O
of	O
"	O
shape	O
"	O
.	O
</s>
<s>
The	O
main	O
tool	O
is	O
persistent	B-Algorithm
homology	I-Algorithm
,	O
an	O
adaptation	O
of	O
homology	O
to	O
point	B-Algorithm
cloud	I-Algorithm
data	I-Algorithm
.	O
</s>
<s>
Persistent	B-Algorithm
homology	I-Algorithm
has	O
been	O
applied	O
to	O
many	O
types	O
of	O
data	O
across	O
many	O
fields	O
.	O
</s>
<s>
The	O
unique	O
features	O
of	O
TDA	O
make	O
it	O
a	O
promising	O
bridge	O
between	O
topology	B-Architecture
and	O
geometry	O
.	O
</s>
<s>
The	O
main	O
insight	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
is	O
to	O
use	O
the	O
information	O
obtained	O
from	O
all	O
parameter	O
values	O
by	O
encoding	O
this	O
huge	O
amount	O
of	O
information	O
into	O
an	O
understandable	O
and	O
easy-to-represent	O
form	O
.	O
</s>
<s>
Precursors	O
to	O
the	O
full	O
concept	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
appeared	O
gradually	O
over	O
time	O
.	O
</s>
<s>
In	O
1990	O
,	O
Patrizio	O
Frosini	O
introduced	O
a	O
pseudo-distance	O
between	O
submanifolds	O
,	O
and	O
later	O
the	O
size	O
function	O
,	O
which	O
on	O
1-dim	O
curves	O
is	O
equivalent	O
to	O
the	O
0th	O
persistent	B-Algorithm
homology	I-Algorithm
.	O
</s>
<s>
introduced	O
the	O
concept	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
together	O
with	O
an	O
efficient	O
algorithm	O
and	O
its	O
visualization	O
as	O
a	O
persistence	O
diagram	O
.	O
</s>
<s>
In	O
algebraic	O
topology	B-Architecture
the	O
persistent	B-Algorithm
homology	I-Algorithm
has	O
emerged	O
through	O
the	O
work	O
of	O
Sergey	O
Barannikov	O
on	O
Morse	O
theory	O
.	O
</s>
<s>
A	O
point	B-Algorithm
cloud	I-Algorithm
is	O
often	O
defined	O
as	O
a	O
finite	O
set	O
of	O
points	O
in	O
some	O
Euclidean	O
space	O
,	O
but	O
may	O
be	O
taken	O
to	O
be	O
any	O
finite	O
metric	O
space	O
.	O
</s>
<s>
The	O
Čech	O
complex	O
of	O
a	O
point	B-Algorithm
cloud	I-Algorithm
is	O
the	O
nerve	O
of	O
the	O
cover	O
of	O
balls	O
of	O
a	O
fixed	O
radius	O
around	O
each	O
point	O
in	O
the	O
cloud	O
.	O
</s>
<s>
A	O
persistence	O
module	O
indexed	O
by	O
is	O
a	O
vector	O
space	O
for	O
each	O
,	O
and	O
a	O
linear	O
map	O
whenever	O
,	O
such	O
that	O
for	O
all	O
and	O
whenever	O
An	O
equivalent	O
definition	O
is	O
a	O
functor	B-Language
from	O
considered	O
as	O
a	O
partially	O
ordered	O
set	O
to	O
the	O
category	O
of	O
vector	O
spaces	O
.	O
</s>
<s>
The	O
persistent	B-Algorithm
homology	I-Algorithm
group	O
of	O
a	O
point	B-Algorithm
cloud	I-Algorithm
is	O
the	O
persistence	O
module	O
defined	O
as	O
,	O
where	O
is	O
the	O
Čech	O
complex	O
of	O
radius	O
of	O
the	O
point	B-Algorithm
cloud	I-Algorithm
and	O
is	O
the	O
homology	O
group	O
.	O
</s>
<s>
A	O
persistence	O
barcode	O
is	O
a	O
multiset	B-Application
of	O
intervals	O
in	O
,	O
and	O
a	O
persistence	O
diagram	O
is	O
a	O
multiset	B-Application
of	O
points	O
in	O
(	O
)	O
.	O
</s>
<s>
The	O
first	O
classification	O
theorem	O
for	O
persistent	B-Algorithm
homology	I-Algorithm
appeared	O
in	O
1994	O
via	O
Barannikov	O
's	O
canonical	O
forms	O
.	O
</s>
<s>
Persistent	B-Algorithm
homology	I-Algorithm
is	O
visualized	O
through	O
a	O
barcode	O
or	O
persistence	O
diagram	O
.	O
</s>
<s>
If	O
is	O
a	O
point	B-Algorithm
cloud	I-Algorithm
,	O
replace	O
with	O
a	O
nested	O
family	O
of	O
simplicial	O
complexes	O
(	O
such	O
as	O
the	O
Čech	O
or	O
Vietoris-Rips	O
complex	O
)	O
.	O
</s>
<s>
This	O
process	O
converts	O
the	O
point	B-Algorithm
cloud	I-Algorithm
into	O
a	O
filtration	O
of	O
simplicial	O
complexes	O
.	O
</s>
<s>
The	O
first	O
algorithm	O
over	O
all	O
fields	O
for	O
persistent	B-Algorithm
homology	I-Algorithm
in	O
algebraic	O
topology	B-Architecture
setting	O
was	O
described	O
by	O
Barannikov	O
through	O
reduction	O
to	O
the	O
canonical	O
form	O
by	O
upper-triangular	O
matrices	O
.	O
</s>
<s>
The	O
algorithm	O
for	O
persistent	B-Algorithm
homology	I-Algorithm
over	O
was	O
given	O
by	O
Edelsbrunner	O
et	O
al	O
.	O
</s>
<s>
Zomorodian	O
and	O
Carlsson	O
gave	O
the	O
practical	O
algorithm	O
to	O
compute	O
persistent	B-Algorithm
homology	I-Algorithm
over	O
all	O
fields	O
.	O
</s>
<s>
Edelsbrunner	O
and	O
Harer	O
's	O
book	O
gives	O
general	O
guidance	O
on	O
computational	B-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
is	O
a	O
Python	O
package	O
dedicated	O
to	O
integrating	O
TDA	O
in	O
the	O
machine	O
learning	O
workflow	O
by	O
means	O
of	O
a	O
scikit-learn	B-Application
API	O
.	O
</s>
<s>
The	O
is	O
specialized	O
for	O
continuous	O
data	O
defined	O
on	O
manifolds	O
of	O
low	O
dimension	O
(	O
1	O
,	O
2	O
or	O
3	O
)	O
,	O
as	O
typically	O
found	O
in	O
scientific	B-Application
visualization	I-Application
.	O
</s>
<s>
Another	O
R	O
package	O
,	O
,	O
implements	O
the	O
Ripser	O
library	O
to	O
calculate	O
persistent	B-Algorithm
homology	I-Algorithm
.	O
</s>
<s>
Many	O
methods	O
have	O
been	O
invented	O
to	O
extract	O
a	O
low-dimensional	O
structure	O
from	O
the	O
data	O
set	O
,	O
such	O
as	O
principal	B-Application
component	I-Application
analysis	I-Application
and	O
multidimensional	O
scaling	O
.	O
</s>
<s>
Thus	O
,	O
the	O
study	O
of	O
visualization	O
of	O
high-dimensional	O
spaces	O
is	O
of	O
central	O
importance	O
to	O
TDA	O
,	O
although	O
it	O
does	O
not	O
necessarily	O
involve	O
the	O
use	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
.	O
</s>
<s>
However	O
,	O
recent	O
attempts	O
have	O
been	O
made	O
to	O
use	O
persistent	B-Algorithm
homology	I-Algorithm
in	O
data	O
visualization	O
.	O
</s>
<s>
Because	O
the	O
topology	B-Architecture
of	O
a	O
finite	O
point	B-Algorithm
cloud	I-Algorithm
is	O
trivial	O
,	O
clustering	O
methods	O
(	O
such	O
as	O
single	B-Algorithm
linkage	I-Algorithm
)	O
are	O
used	O
to	O
produce	O
the	O
analogue	O
of	O
connected	O
sets	O
in	O
the	O
preimage	O
when	O
MAPPER	O
is	O
applied	O
to	O
actual	O
data	O
.	O
</s>
<s>
Persistent	O
space	O
,	O
a	O
generalization	O
of	O
persistent	O
diagram	O
,	O
is	O
defined	O
as	O
the	O
multiset	B-Application
of	O
all	O
points	O
with	O
multiplicity	O
larger	O
than	O
0	O
and	O
the	O
diagonal	O
.	O
</s>
<s>
is	O
trying	O
to	O
give	O
geometric	O
interpretation	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
,	O
which	O
might	O
provide	O
insights	O
on	O
how	O
to	O
combine	O
machine	O
learning	O
theory	O
with	O
topological	B-Algorithm
data	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
It	O
's	O
natural	O
to	O
extend	O
persistence	O
homology	O
to	O
other	O
basic	O
concepts	O
in	O
algebraic	O
topology	B-Architecture
,	O
such	O
as	O
cohomology	O
and	O
relative	O
homology/cohomology	O
.	O
</s>
<s>
Category	O
theory	O
is	O
the	O
language	O
of	O
modern	O
algebra	O
,	O
and	O
has	O
been	O
widely	O
used	O
in	O
the	O
study	O
of	O
algebraic	O
geometry	O
and	O
topology	B-Architecture
.	O
</s>
<s>
Following	O
the	O
notations	O
made	O
in	O
Bubenik	O
et	O
al.	O
,	O
the	O
indexing	O
category	O
is	O
any	O
preordered	O
set	O
(	O
not	O
necessarily	O
or	O
)	O
,	O
the	O
target	O
category	O
is	O
any	O
category	O
(	O
instead	O
of	O
the	O
commonly	O
used	O
)	O
,	O
and	O
functors	B-Language
are	O
called	O
generalized	O
persistence	O
modules	O
in	O
,	O
over	O
.	O
</s>
<s>
Another	O
example	O
is	O
the	O
relationship	O
between	O
the	O
construction	O
of	O
different	O
complexes	O
from	O
point	B-Algorithm
clouds	I-Algorithm
.	O
</s>
<s>
Justin	O
Curry	O
regards	O
level	O
set	O
persistence	O
as	O
the	O
study	O
of	O
fibers	B-Algorithm
of	O
continuous	O
functions	O
.	O
</s>
<s>
have	O
abstracted	O
the	O
definition	O
to	O
that	O
between	O
functors	B-Language
when	O
is	O
equipped	O
with	O
a	O
sublinear	O
projection	O
or	O
superlinear	O
family	O
,	O
in	O
which	O
still	O
remains	O
a	O
pseudometric	O
.	O
</s>
<s>
Let	O
be	O
a	O
functor	B-Language
between	O
arbitrary	O
categories	O
.	O
</s>
<s>
Then	O
for	O
any	O
two	O
functors	B-Language
,	O
we	O
have	O
.	O
</s>
<s>
Persistent	B-Algorithm
homology	I-Algorithm
has	O
no	O
inherent	O
mechanism	O
to	O
distinguish	O
between	O
low-probability	O
features	O
and	O
high-probability	O
features	O
.	O
</s>
<s>
One	O
way	O
to	O
apply	O
statistics	O
to	O
topological	B-Algorithm
data	I-Algorithm
analysis	I-Algorithm
is	O
to	O
study	O
the	O
statistical	O
properties	O
of	O
topological	O
features	O
of	O
point	B-Algorithm
clouds	I-Algorithm
.	O
</s>
<s>
The	O
study	O
of	O
random	O
simplicial	O
complexes	O
offers	O
some	O
insight	O
into	O
statistical	O
topology	B-Architecture
.	O
</s>
<s>
Besides	O
the	O
obvious	O
need	O
for	O
algebra	O
and	O
topology	B-Architecture
,	O
partial	O
differential	O
equations	O
,	O
algebraic	O
geometry	O
,	O
representation	O
theory	O
,	O
statistics	O
,	O
combinatorics	O
,	O
and	O
Riemannian	O
geometry	O
have	O
all	O
found	O
use	O
in	O
TDA	O
.	O
</s>
<s>
Topology	B-Architecture
is	O
considered	O
to	O
be	O
very	O
soft	O
since	O
many	O
concepts	O
are	O
invariant	O
under	O
homotopy	O
.	O
</s>
<s>
However	O
,	O
persistent	O
topology	B-Architecture
is	O
able	O
to	O
record	O
the	O
birth	O
(	O
appearance	O
)	O
and	O
death	O
(	O
disappearance	O
)	O
of	O
topological	O
features	O
,	O
thus	O
extra	O
geometric	O
information	O
is	O
embedded	O
in	O
it	O
.	O
</s>
<s>
Topological	B-Algorithm
data	I-Algorithm
analysis	I-Algorithm
and	O
persistent	B-Algorithm
homology	I-Algorithm
have	O
had	O
impacts	O
on	O
Morse	O
theory	O
.	O
</s>
<s>
Some	O
work	O
in	O
persistent	B-Algorithm
homology	I-Algorithm
has	O
extended	O
results	O
about	O
Morse	O
functions	O
to	O
tame	O
functions	O
or	O
,	O
even	O
to	O
continuous	O
functions	O
.	O
</s>
<s>
A	O
forgotten	O
result	O
of	O
R	O
.	O
Deheuvels	O
long	O
before	O
the	O
invention	O
of	O
persistent	B-Algorithm
homology	I-Algorithm
extends	O
Morse	O
theory	O
to	O
all	O
continuous	O
functions	O
.	O
</s>
<s>
Persistent	B-Algorithm
homology	I-Algorithm
is	O
closely	O
related	O
to	O
spectral	O
sequences	O
.	O
</s>
