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11. TOPOLOGICAL PERSISTENCE
Filtered complexes arise naturally in many situations. e simplest example of filtration is
the
age filtration
: the complex
K
is filtered by giving an ordering
0
;
1
;:::;
m
to its simplexes
and by defining the sequence of its subcomplexes
K
i
as
K
i
Df
j
2KW0jig
. In other
words the complex grows from
K
0
Df
0
g
adding each simplex one by one according to the
given order.
A filtered complex also occurs when some space (e.g., a curve or a surface) is known only
through a finite sample
X
of its points. Since the knowledge of the original space is necessarily
imprecise, a multi-scale approach may be suited to describe the topology of the underlying space.
e idea is to construct, for a real number
> 0
, an abstract simplicial complex
R
.X/
, called
the
Rips complex
, whose abstract
k
-simplexes are exactly the subsets
fx
0
;x
1
;:::;x
k
g
of
X
such
that
d.x
i
;x
j
/
for all pairs
x
i
;x
j
with
0i;jk
. Whenever
<
0
, there is an inclusion
R
.X/!R
0
.X/
that reveals a growing complex.
Given a filtered complex, its topological attributes change through the filtration, since new
components appear or connect to the old ones, tunnels are created and closed off, voids are en-
closed and filled in, etc. In particular, as for
0
-homology, each homology class corresponds to
a connected component, and a homology class is born when a point is added, forming a new
connected component, thus being a
0
-cycle. A homology class dies when two points belong-
ing to different connected components, i.e., they belong to two different
0
-cycles, are connected
by a
1
-chain, thus becoming a boundary. As an example, consider the filtered complex of Fig-
ure
11.1
(top): one
0
-homology class originates at
K
0
, two other homology classes arise at
K
1
;
at
K
2
, a new homology class is created while one of the classes originated at
K
1
dies, since it
is merged to the class born at
K
0
; at
K
3
another class disappears, and the same happens at
K
4
,
where we are left with just one class that survives forever. As for
1
-homology, a homology class is
born when a
1
-chain is added, forming a
1
-cycle (for instance, a
1
-simplex is added, completing a
circle), while it dies when a
2
-chain is added so that the
1
-cycle becomes a boundary (for instance,
a
2
-simplex fills a circle). In the example of Figure
11.1
(top), a homology class is born at
K
3
,
another one at
K
4
, then at
K
5
the homology class born at
K
3
dies and at
K
6
also the homology
class born at
K
4
dies, so that no
1
-cycle survives any longer. e argument goes on similarly for
higher degree homology. Persistent homology algebraically captures this process of the birth and
death of homology classes.
More formally, given a filtered simplicial complex
fK
i
g
iD0;:::;n
, the
j
- persistent
k
-th ho-
mology group
of
K
i
can be defined as a group isomorphic to the image of the homomorphism
i;j
k
WH
k
.K
i
/!H
k
.K
iCj
/
induced by the inclusion of
K
i
into
K
iCj
. Also, the
j
-persistent ho-
mology group
of
K
i
counts how many homology classes of
K
i
still survive in
K
iCj
. Persistence
represents the life-time of cycles in the growing filtration.
e persistent homology of a filtered complex can be represented by a set of intervals, called
persistence intervals (briefly
-interval is a pair
.i;j/
,
with
i;j2Z[fC1g
and
0i < j
, such that there exists a cycle that is completed at level
i
of
P
-intervals
), as in Figure
11.1
(bottom): a
P
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