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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