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C H A P T E R
11
Topological Persistence
Persistent homology couples notions of algebraic topology (see chapter
7
) with Morse theoretic
reasoning: as we have seen in chapter
8
, Morse theory explores the topological attributes of an
object in an evolutionary context. is concept of evolution has been rethought in [
72
], where
the authors introduced
persistence
, a technique which grows a space incrementally and analyzes
the placement of topological events within the history of this growth: for example, the
birth
of a
connected component and its
death
when it merges into another component, the birth of a hole
and its death after it is filled (section
11.1
).
e aim was to furnish a scale to assess the relevance of topological attributes, under the
assumption that
longevity
is equivalent to
significance
. In other words, a significant topological
attribute must have a long lifetime in a growing complex, whereas noise and details are short-lived.
In this way, one is able to distinguish the essential features from the fine details. e lifespan of
topological attributes is encoded in a simple and compact representation called
persistencediagram
,
which we describe in section
11.2
. eir multidimensional analogues, namely
persistence spaces
, are
the subject of section
11.3
.
We notice that similarly to many concepts in mathematics, though the term
persistence
was
introduced in [
71
] in 2000, the concept has a historical root system that comprises the work of
Patrizio Frosini and collaborators, who in the 1990s introduced size functions [
81
], which are
equivalent to 0-persistent homology, and the study of the homology of sample spaces by Vanessa
Robins [
170
] that dates back to 1999.
Understanding this chapter requires the notions about simplicial complexes and homology
introduced in chapter
7
.
11.1 BASIC CONCEPTS
e first concept related to persistent homology theory is that of a filtered complex, that is a
complex equipped with a filtration.
A
filtration
of a complex is a nested sequence of subcomplexes that ends with the com-
plex itself. Formally, a complex
K
is filtered by a filtration
fK
i
g
iD0;:::;n
if
K
n
DK
and
K
i
is a
subcomplex of
K
iC1
for each
iD0;:::;n1
.
An example of filtered complex is given in Figure
11.1
(top). Since the sequence of sub-
complexes
K
i
is nested, one can think of
K
as a complex that grows from an initial state
K
0
to a
final state
K
n
DK
. erefore
K
is often referred to as a growing complex.
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