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The persistence space in multidimensional persistent homology

Profile image of Claudia LandiClaudia Landi

2013, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

https://doi.org/10.1007/978-3-642-37067-0-16
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12 pages

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Abstract

Multidimensional persistent modules do not admit a concise representation analogous to that provided by persistence diagrams for real-valued functions. However, there is no obstruction for multidimensional persistent Betti numbers to admit one. Therefore, it is reasonable to look for a generalization of persistence diagrams concerning those properties that are related only to persistent Betti numbers. In this paper, the persistence space of a vector-valued continuous function is introduced to generalize the concept of persistence diagram in this sense. Furthermore, it is presented a method to visualize topological features of a shape via persistence spaces. Finally, it is shown that this method is resistant to perturbations of the input data.

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References (27)

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Multidimensional persistent homology is stable
Claudia Landi, Patrizio Frosini

Arxiv preprint arXiv: …, 2009

Multidimensional persistence studies topological features of shapes by analyzing the lower level sets of vector-valued functions. The rank invariant completely determines the multidimensional analogue of persistent homology groups. We prove that multidimensional rank invariants are stable with respect to function perturbations. More precisely, we construct a distance between rank invariants such that small changes of the function imply only small changes of the rank invariant. This result can be obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.

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Abstract: The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, ie for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out.

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Betti numbers in multidimensional persistent homology are stable functions
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Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions. As is well known, in the case of scalar-valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, i.e. the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector-valued filtering functions, we can consider the multidimensional analogue of persistent Betti numbers. Varying the lower level sets, we get that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non-negative integers. In this paper we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector-valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector-valued filtering functions, and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalarvalued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers we obtain a lower bound for the natural pseudo-distance.

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We consider generic curves in R 2 , i.e. generic C 1 functions f : S 1 → R 2 . We analyze these curves through the persistent homology groups of a filtration induced on S 1 by f . In particular, we consider the question whether these persistent homology groups uniquely characterize f , at least up to re-parameterizations of S 1 . We give a partially positive answer to this question. More precisely, we prove that f = g • h, where h : S 1 → S 1 is a C 1 -diffeomorphism, if and only if the persistent homology groups of s • f and s • g coincide, for every s belonging to the group Σ 2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s ∈ Σ 2 , the persistent Betti numbers functions of s • f and s • g are close to each other, with respect to a suitable distance.

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