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Reparametrization invariant norms [article]

Patrizio Frosini, Claudia Landi
2007 arXiv   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
more » ... 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.
arXiv:math/0702094v1 fatcat:m73eudckvvbnfjsqf5crqbuani

Filtrations induced by continuous functions [article]

Barbara Di Fabio, Patrizio Frosini
2013 arXiv   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
more » ... 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.
arXiv:1304.1268v1 fatcat:j6jago5uanblddrmxitg7275gq

Stability in multidimensional Size Theory [article]

Andrea Cerri, Patrizio Frosini, Claudia Landi
2006 arXiv   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

Does intelligence imply contradiction? [article]

Patrizio Frosini
2008 arXiv   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
more » ... 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.
arXiv:0801.0232v2 fatcat:7i7mx7d5wjhpfnctgc3ktoxsqa

Stable comparison of multidimensional persistent homology groups with torsion [article]

Patrizio Frosini
2010 arXiv   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
more » ... 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.
arXiv:1012.4169v1 fatcat:5fsdcaewwnhmvdfilg6dnv3d4m

Multidimensional persistent homology is stable [article]

Andrea Cerri, Barbara Di Fabio, Massimo Ferri, Patrizio Frosini, Claudia Landi
2009 arXiv   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
more » ... obtained by assuming the function to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence.
arXiv:0908.0064v1 fatcat:mxcfsug7nzfinhs4n4ezytnca4

Position paper: Towards an observer-oriented theory of shape comparison [article]

Patrizio Frosini
2016 arXiv   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
more » ... reasons that justify its possible use for shape comparison.
arXiv:1603.02008v1 fatcat:vtuvfpo6ivhztbopiebspudrmm

The coherent matching distance in 2D persistent homology [article]

Andrea Cerri, Marc Ethier, Patrizio Frosini
2016 arXiv   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
more » ... 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.
arXiv:1603.03886v1 fatcat:zqytvqkmtrfpvejdorhhodl4dq

Natural pseudo-distances between closed curves

Pietro Donatini, Patrizio Frosini
2009 Forum mathematicum  
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
more » ... 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
doi:10.1515/forum.2009.049 fatcat:irgzhqp2szfyjg3gaqooxfm3h4

The Persistent Homotopy Type Distance [article]

Patrizio Frosini, Claudia Landi, Facundo Memoli
2018 arXiv   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
more » ... 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.
arXiv:1702.07893v2 fatcat:dbnpm7oj3fe4pftr6arzbbpgoq

ADVANCES IN MULTIDIMENSIONAL SIZE THEORY

Andrea Cerri, Patrizio Frosini
2011 Image Analysis and Stereology  
For further details about Size Theory, the reader is referred to 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) .  ... 
doi:10.5566/ias.v29.p19-26 fatcat:ythxqwg5ubfxnkto4u2yv2umqy

G-invariant persistent homology

Patrizio Frosini
2014 Mathematical methods in the applied sciences  
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
more » ... 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.
doi:10.1002/mma.3139 fatcat:l4b3ylew45aqfoebwgxn7q3iay

The persistent homotopy type distance

Patrizio Frosini, Claudia Landi, Facundo Mémoli
2019 Homology, Homotopy and Applications  
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
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.
doi:10.4310/hha.2019.v21.n2.a13 fatcat:qsxdthg2hjgnzfscdnvvccpz2m

Necessary Conditions for Discontinuities of Multidimensional Size Functions [article]

Andrea Cerri, Patrizio Frosini
2009 arXiv   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

Natural pseudodistances between closed surfaces

Patrizio Frosini, Pietro Donatini
2007 Journal of the European Mathematical Society (Print)  
Finally, 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).  ... 
doi:10.4171/jems/82 fatcat:cwtzs4mrkffxrhua35mgkaelx4
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