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Reparametrization invariant norms
[article]

2007
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arXiv
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pre-print

This paper explores the concept of reparametrization invariant norm (RPI-norm), that is any norm invariant under composition with diffeomorphisms. We prove the existence of an infinite family of RPI-norms, called standard RPI-norms, for which we exhibit both an integral and a discrete characterization. Our main result states that, for every one-time differentiable piecewise monotone function with compact support, its standard RPI-norms allow us to compute the value of any other RPI-norm of the

arXiv:math/0702094v1
fatcat:m73eudckvvbnfjsqf5crqbuani
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... ame function. This is proved by using the standard RPI-norms in order to reconstruct the function up to reparametrization and an arbitrarily small error with respect to the total variation norm.##
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Filtrations induced by continuous functions
[article]

2013
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arXiv
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pre-print

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to ^n. A natural question arises, whether these approaches are equivalent or not. In this paper we study this problem and prove that, while the answer to the previous question is negative in the general case, the approach by continuous functions is not restrictive with respect to the other, provided that some natural stability and

arXiv:1304.1268v1
fatcat:j6jago5uanblddrmxitg7275gq
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... eteness assumptions are made. In particular, we show that every compact and stable 1-dimensional filtration of a compact metric space is induced by a continuous function. Moreover, we extend the previous result to the case of multi-dimensional filtrations, requiring that our filtration is also complete. Three examples show that we cannot drop the assumptions about stability and completeness. Consequences of our results on the definition of a distance between filtrations are finally discussed.##
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Stability in multidimensional Size Theory
[article]

2006
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arXiv
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pre-print

This paper proves that in Size Theory the comparison of multidimensional size functions can be reduced to the 1-dimensional case by a suitable change of variables. Indeed, we show that a foliation in half-planes can be given, such that the restriction of a multidimensional size function to each of these half-planes turns out to be a classical size function in two scalar variables. This leads to the definition of a new distance between multidimensional size functions, and to the proof of their stability with respect to that distance.

arXiv:cs/0608009v1
fatcat:ecocmu37qrhvfkslhcl7y43q3a
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Does intelligence imply contradiction?
[article]

2008
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arXiv
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pre-print

Contradiction is often seen as a defect of intelligent systems and a dangerous limitation on efficiency. In this paper we raise the question of whether, on the contrary, it could be considered a key tool in increasing intelligence in biological structures. A possible way of answering this question in a mathematical context is shown, formulating a proposition that suggests a link between intelligence and contradiction. A concrete approach is presented in the well-defined setting of cellular

arXiv:0801.0232v2
fatcat:7i7mx7d5wjhpfnctgc3ktoxsqa
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... ata. Here we define the models of "observer", "entity", "environment", "intelligence" and "contradiction". These definitions, which roughly correspond to the common meaning of these words, allow us to deduce a simple but strong result about these concepts in an unbiased, mathematical manner. Evidence for a real-world counterpart to the demonstrated formal link between intelligence and contradiction is provided by three computational experiments.##
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Stable comparison of multidimensional persistent homology groups with torsion
[article]

2010
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arXiv
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pre-print

The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of

arXiv:1012.4169v1
fatcat:5fsdcaewwnhmvdfilg6dnv3d4m
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... e new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between d_T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.##
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Multidimensional persistent homology is stable
[article]

2009
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arXiv
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pre-print

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

arXiv:0908.0064v1
fatcat:mxcfsug7nzfinhs4n4ezytnca4
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... 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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Position paper: Towards an observer-oriented theory of shape comparison
[article]

2016
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arXiv
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pre-print

In this position paper we suggest a possible metric approach to shape comparison that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions represent the set of data that are accessible to the observer, while the operators describe the way the observer elaborates the data and enclose the invariance that he/she associates with them. We expose this model and illustrate some theoretical

arXiv:1603.02008v1
fatcat:vtuvfpo6ivhztbopiebspudrmm
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... reasons that justify its possible use for shape comparison.##
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The coherent matching distance in 2D persistent homology
[article]

2016
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arXiv
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pre-print

Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link between the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological

arXiv:1603.03886v1
fatcat:zqytvqkmtrfpvejdorhhodl4dq
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... ormation is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties. In this paper we introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations we take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the above filtrations can induce different matchings between the associated persistent diagrams. In our paper we prove that the coherent 2D matching distance is well-defined and stable.##
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Natural pseudo-distances between closed curves

2009
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Forum mathematicum
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Let us consider two closed curves M, N of class C 1 and two functions ϕ : M → IR, ψ : N → IR of class C 1 , called measuring functions. The natural pseudo-distance d between the pairs (M, ϕ), (N , ψ) is defined as the infimum of Θ(f ) def = max P ∈M |ϕ(P ) − ψ(f (P ))|, as f varies in the set of all homeomorphisms from M onto N . The problem of finding the possible values for d naturally arises. In this paper we prove that under appropriate hypotheses the natural pseudo-distance equals either

doi:10.1515/forum.2009.049
fatcat:irgzhqp2szfyjg3gaqooxfm3h4
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... 1 −c 2 | or 1 2 |c 1 −c 2 |, where c 1 and c 2 are two suitable critical values of the measuring functions. This equality shows that the relations between the natural pseudo-distance and the critical values of the measuring functions previously obtained in higher dimensions can be made stronger in the particular case of closed curves. Moreover, the examples we give in this paper show that our result cannot be further improved, and therefore it completely solves the problem of determining the possible values for d in the 1##
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The Persistent Homotopy Type Distance
[article]

2018
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arXiv
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pre-print

We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that dHT still provides

arXiv:1702.07893v2
fatcat:dbnpm7oj3fe4pftr6arzbbpgoq
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... an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the L-infty distance and the natural pseudo-distance dNP. From a different standpoint, we prove that dHT extends the L-infty distance and dNP in two ways. First, we show that, appropriately restricting the category of objects to which dHT applies, it can be made to coincide with the other two distances. Finally, we show that dHT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.##
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ADVANCES IN MULTIDIMENSIONAL SIZE THEORY

2011
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Image Analysis and Stereology
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For further details about Size Theory, the reader is referred to

doi:10.5566/ias.v29.p19-26
fatcat:ythxqwg5ubfxnkto4u2yv2umqy
*Frosini*and Mulazzani (1999) ; Biasotti et al. (2007; 2008a; . ... Indeed, in Cerri and*Frosini*(2008) it is shown that a correspondence exists between the discontinuity points of ℓ (M ,F) and the ones of ℓ (M , ϕ) . ...##
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G-invariant persistent homology

2014
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Mathematical methods in the applied sciences
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Classical persistent homology is not tailored to study the action of transformation groups different from the group Homeo(X) of all self-homeomorphisms of a topological space X. In order to obtain better lower bounds for the natural pseudo-distance d G associated with a group G ⊂ Homeo(X), we need to adapt persistent homology and consider G-invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under

doi:10.1002/mma.3139
fatcat:l4b3ylew45aqfoebwgxn7q3iay
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... he action of G. In this paper we formalize this idea, and prove the stability of G-invariant persistent homology with respect to the natural pseudo-distance d G . We also show how G-invariant persistent homology could be used in applications concerning shape comparison.##
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The persistent homotopy type distance

2019
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Homology, Homotopy and Applications
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We introduce the persistent homotopy type distance d HT to compare two real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of d HT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopy equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that d HT still

doi:10.4310/hha.2019.v21.n2.a13
fatcat:qsxdthg2hjgnzfscdnvvccpz2m
## more »

... s an upper bound for the bottleneck distance between the persistence diagrams of the intervening functions. Moreover, because homotopy equivalences are weaker than homeomorphisms, this implies a lifting of the standard stability results provided by the L ∞ distance and the natural pseudo-distance d NP . From a different standpoint, we prove that d HT extends the L ∞ distance and d NP in two ways. First, we show that, appropriately restricting the category of objects to which d HT applies, it can be made to coincide with the other two distances. Finally, we show that d HT has an interpretation in terms of interleavings that naturally places it in the family of distances used in persistence theory.##
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Necessary Conditions for Discontinuities of Multidimensional Size Functions
[article]

2009
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arXiv
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pre-print

Some new results about multidimensional Topological Persistence are presented, proving that the discontinuity points of a k-dimensional size function are necessarily related to the pseudocritical or special values of the associated measuring function.

arXiv:0811.1868v2
fatcat:nwxndmilgnf73nadu4xpmzyqzq
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Natural pseudodistances between closed surfaces

2007
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Journal of the European Mathematical Society (Print)
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Finally,

doi:10.4171/jems/82
fatcat:cwtzs4mrkffxrhua35mgkaelx4
*Frosini*wishes to thank I. Fossati, F. Battiato and A. Branduardi for their indispensable support. This work was partially supported by MIUR (Italy), ARCES (Italy) and INdAM-GNSAGA (Italy). ...
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