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The theory of multidimensional persistence

Gunnar Carlsson, Afra Zomorodian
2007 Proceedings of the twenty-third annual symposium on Computational geometry - SCG '07  
In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence.  ...  Persistent homology captures the topology of a filtration -a one-parameter family of increasing spaces -in terms of a complete discrete invariant.  ...  Introduction In this paper, we introduce the theory of multidimensional persistence, the extension of the concept of persistent homology [7, 15] .  ... 
doi:10.1145/1247069.1247105 dblp:conf/compgeom/CarlssonZ07 fatcat:a6yuffwrt5ghvbenrc4gxrblm4

The Theory of Multidimensional Persistence

Gunnar Carlsson, Afra Zomorodian
2009 Discrete & Computational Geometry  
In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence.  ...  Persistent homology captures the topology of a filtration-a one-parameter family of increasing spaces-in terms of a complete discrete invariant.  ...  Introduction In this paper, we introduce the theory of multidimensional persistence, an extension of the concept of persistent homology [9, 21] .  ... 
doi:10.1007/s00454-009-9176-0 fatcat:lfsthnmwlbgi5cgu5rohrazvsy

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

Michael Lesnick
2015 Foundations of Computational Mathematics  
In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results.  ...  The definitions of ϵ-interleavings and d_I generalize readily to multidimensional persistence modules.  ...  In our many conversations about multidimensional persistent homology, Gunnar has impressed on me the value of bringing the machinery of commutative algebra to bear on the study of multidimensional persistence  ... 
doi:10.1007/s10208-015-9255-y fatcat:iqvqryci3bccthxeigx3pt5kxe

Topology, Computation and Data Analysis (Dagstuhl Seminar 17292)

Hamish Carr, Michael Kerber, Bei Wang, Marc Herbstritt
2018 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 17292 "Topology, Computation and Data Analysis".  ...  This seminar was the first of its kind in bringing together researchers with mathematical and computational backgrounds in addressing emerging directions within computational topology for data analysis  ...  Multidimensional persistent homology. The second area of active research, both mathematically and computationally, is the extension of unidimensional persistence to multidimensional persistence.  ... 
doi:10.4230/dagrep.7.7.88 dblp:journals/dagstuhl-reports/CarrKW17 fatcat:sxtm3wctuzen7korf5sdbqhkr4

Multidimensional persistence in biomolecular data [article]

Kelin Xia, Guo-Wei Wei
2014 arXiv   pre-print
We introduce two families of multidimensional persistence, namely pseudo-multidimensional persistence and multiscale multidimensional persistence.  ...  The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.  ...  The authors acknowledge the Mathematical Biosciences Institute for hosting valuable workshops.  ... 
arXiv:1412.7679v1 fatcat:miqlwa5wxfaktmtcg4aort4vha

Perfectionism, personality, and affective experiences: New insights from revised Reinforcement Sensitivity Theory

Joachim Stoeber, Philip J. Corr
2015 Personality and Individual Differences  
The present study explored the relationships between three forms of perfectionism (self-oriented, other-oriented, socially prescribed) and components of the revised Reinforcement Sensitivity Theory of  ...  with BAS goaldrive persistence (and was unrelated to the FFFS).  ...  Multidimensional perfectionism and reinforcement sensitivity The Reinforcement Sensitivity Theory (RST) is a prominent neuropsychological theory of personality explaining the role of individual differences  ... 
doi:10.1016/j.paid.2015.06.045 fatcat:4zljkbec3fbo3o7o5hlxg66qhq

Multidimensional persistence in biomolecular data

Kelin Xia, Guo-Wei Wei
2015 Journal of Computational Chemistry  
We introduce two families of multidimensional persistence, namely pseudomultidimensional persistence and multiscale multidimensional persistence.  ...  The multiscale multidimensional persistent homology reveals relative local features in Betti-0 invariants and the relatively global characteristics of Betti-1 and Betti-2 invariants.  ...  Keywords: multidimensional persistence Á multifiltration Á anisotropic filtration Á multiscale persistence Á protein folding Á protein flexibility Á topological denoising  ... 
doi:10.1002/jcc.23953 pmid:26032339 pmcid:PMC4485576 fatcat:w3f77xcthrawthbhqjxljaxsfe

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.  ...  Work performed within the activity of ARCES "E. De Castro", University of Bologna, under the auspices of INdAM-GNSAGA.  ... 
arXiv:0811.1868v2 fatcat:nwxndmilgnf73nadu4xpmzyqzq

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.  ...  The authors thank Francesca Cagliari (University of Bologna) and Marco Grandis (University of Genoa) for their helpful advice. However, the authors are solely responsible for any errors.  ... 
arXiv:0908.0064v1 fatcat:mxcfsug7nzfinhs4n4ezytnca4

Betti numbers in multidimensional persistent homology are stable functions

Andrea Cerri, Barbara Di Fabio, Massimo Ferri, Patrizio Frosini, Claudia Landi
2013 Mathematical methods in the applied sciences  
Finally, from the stability of multidimensional persistent Betti numbers we obtain a lower bound for the natural pseudo-distance.  ...  Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions.  ...  At the time of submission, the first author was visiting the Pattern Recognition and Image Processing Group, Institute of Computer Graphics and Algorithms, Faculty of Informatics, Vienna University of  ... 
doi:10.1002/mma.2704 fatcat:dy4iqplm5zgazp3lkbkfnarm2u

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.  ...  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.  ...  This paper is dedicated to the memory of Marco Gori.  ... 
arXiv:cs/0608009v1 fatcat:ecocmu37qrhvfkslhcl7y43q3a

One-Dimensional Reduction of Multidimensional Persistent Homology [article]

F. Cagliari, B. Di Fabio, M. Ferri
2008 arXiv   pre-print
This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.  ...  A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one.  ...  After recalling some basic notions about multidimensional size functions and 1-dimensional persistent homology in Section 2, we adapt the arguments of [1] to multidimensional persistent homology in Section  ... 
arXiv:math/0702713v2 fatcat:lzira6hxuvd37lsgflabzitbh4

One-dimensional reduction of multidimensional persistent homology

Francesca Cagliari, Barbara Di Fabio, Massimo Ferri
2010 Proceedings of the American Mathematical Society  
This leads to a stable distance for multidimensional persistent homology. Some reflections on the i-essentiality of homological critical values conclude the paper.  ...  A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one.  ...  License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use  ... 
doi:10.1090/s0002-9939-10-10312-8 fatcat:q366dodftfdrtkawiqo6j3iwua

Stability of multidimensional persistent homology with respect to domain perturbations [article]

Patrizio Frosini, Claudia Landi
2010 arXiv   pre-print
We show that by encoding sets using the distance function, the multidimensional matching distance between rank invariants of persistent homology groups is always upperly bounded by the Hausdorff distance  ...  Also in these cases we present results stating that the multidimensional matching distance between rank invariants of persistent homology groups is upperly bounded by these distances.  ...  The stability of multidimensional rank invariants is quite an important issue in persistent homology theory and its applications because the lack of stability would make this invariant useless, every data  ... 
arXiv:1001.1078v2 fatcat:iddr47qfabhitleqtm7nc3ruca

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.  ...  Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion.  ...  The author thanks Sara, and all his friends in the "Coniglietti" Band.  ... 
arXiv:1012.4169v1 fatcat:5fsdcaewwnhmvdfilg6dnv3d4m
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