Graphics Reference
In-Depth Information
Definition 11.1
Multiplicity.
e
multiplicity
f
.u;v/
of
.u;v/2
C
is the finite, non-
negative number given by
min
">0
uC"<v"
f
.uC";v"/
f
.u";v"/C
C
f
.u";vC"/
f
.uC";vC"/:
e
multiplicity
f
.u;1/
of
.u;1/
is the finite, non-negative number given by
">0;uC"<v
f
.uC";v/
f
.u";v/:
min
Definition 11.2
Persistence diagram.
e persistence diagram Dgm
.f /
is the multiset of all
points
p2
such that
f
.p/ > 0
, considered with their multiplicity, union the points of
,
considered with infinite multiplicity.
e computation of persistence diagrams of filtered simplicial complexes presented in [
216
]
exploits parallels between algebraic relationships and matrix representations. e literature also
offers several techniques to speed up homology computation [
88
], which can be employed in the
context of persistence. ey generally focus on the reduction of the size of the input complex
using combinatorial operations which preserve the homology.
11.3 PERSISTENCE SPACES
A recent advance in persistent homology theory is the generalization to filtrations obtained
through a multi-variate function taking value in
R
n
,
n > 1
. Indeed, it is often the case that the
data under study (e.g., scientific data from physical or medical studies) are characterized by a large
number of measurements, which can be modeled as multi-variate functions.
When
f
is vector-valued, i.e.,
fD.f
i
/WX!R
n
, the definition of the multidimensional
analogue of persistent homology groups and Betti numbers is straightforward. For
uD.u
i
/;vD
.v
i
/2R
n
, we say that
uv
(resp.
uv
,
uv
) iff
u
i
< v
i
(resp.
u
i
v
i
,
u
i
> v
i
) for every
iD1;:::;n
. Given
uv
, the
multidimensional
k
th persistent homology group
of
.X;f /
at
.u;v/
is the image of the homomorphism
H
k
.X
u
/!H
k
.X
v
/
induced in homology by the inclusion
of
H
k
.X
u
/
into
H
k
.X
v
/
. Its rank, still denoted by
f
.u;v/
, is called a
multidimensional persistent
Betti number
.
e multidimensional counterpart of the Definitions
11.1
and
11.2
are as follows [
43
]. For
every
.u;v/2
Df.u;v/2R
n
R
n
Wuvg
and
e2R
n
with
e0
and
uCeve
, we
consider the number
C
n
e
f
.u;v/D
f
.uCe;ve/
f
.ue;ve/C
C
f
.ue;vCe/
f
.uCe;vCe/:
(11.1)
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