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(2002)
Algebra, revised third edition
Patrizio Frosini (2010)
Stable Comparison of Multidimensional Persistent Homology Groups with TorsionActa Applicandae Mathematicae, 124
G. Carlsson, T. Ishkhanov, V. Silva, A. Zomorodian (2007)
On the Local Behavior of Spaces of Natural ImagesInternational Journal of Computer Vision, 76
Chunyuan Li, M. Ovsjanikov, F. Chazal (2014)
Persistence-Based Structural Recognition2014 IEEE Conference on Computer Vision and Pattern Recognition
M. Atiyah (1956)
On the Krull-Schmidt theorem with application to sheavesBulletin de la Société Mathématique de France, 84
L. Wasserman (2004)
All of Statistics: A Concise Course in Statistical Inference
Peter Bubenik, Jonathan Scott (2012)
Categorification of Persistent HomologyDiscrete & Computational Geometry, 51
F. Chazal, L. Guibas, S. Oudot, P. Skraba (2009)
Analysis of scalar fields over point cloud data
Günter Rote, Gert Vegter (2007)
Effective Computational Geometry for Curves and Surfaces Chapter 7 Computational Topology : An Introduction
A Hatcher (2002)
Algebraic topology
F. Chazal, D. Cohen-Steiner, M. Glisse, L. Guibas, S. Oudot (2009)
Proximity of persistence modules and their diagrams
J. Curry (2013)
Sheaves, Cosheaves and ApplicationsarXiv: Algebraic Topology
Aaron Adcock, D. Rubin, G. Carlsson (2012)
Classification of hepatic lesions using the matching metricComput. Vis. Image Underst., 121
S. Biasotti, A. Cerri, Patrizio Frosini, D. Giorgi, C. Landi (2008)
Multidimensional Size Functions for Shape ComparisonJournal of Mathematical Imaging and Vision, 32
Noname manuscript No. (will be inserted by the editor) Interleaving Distance between Merge Trees
D Cohen-Steiner, H Edelsbrunner, J Harer, Y Mileyko (2010)
Lipschitz functions have $$L_p$$ L p -stable persistenceFoundations of Computational Mathematics, 10
V. Silva, E. Munch, A. Patel (2015)
Categorified Reeb GraphsDiscrete & Computational Geometry, 55
M. d'Amico, Patrizio Frosini, C. Landi (2008)
Natural Pseudo-Distance and Optimal Matching between Reduced Size FunctionsActa Applicandae Mathematicae, 109
H Derksen, J Weyman (2005)
Quiver representationsNotices of the AMS, 52
G. Carlsson (2014)
Topological pattern recognition for point cloud data*Acta Numerica, 23
A. Zomorodian, G. Carlsson (2004)
Computing Persistent HomologyDiscrete & Computational Geometry, 33
Cary Webb (1985)
Decomposition of graded modules, 94
D Morozov, K Beketayev, G Weber (2013)
Interleaving distance between merge treesDiscrete and Computational Geometry, 49
F. Chazal, W. Crawley-Boevey, V. Silva (2014)
The observable structure of persistence modulesarXiv: Representation Theory
D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Yuriy Mileyko (2010)
Lipschitz Functions Have Lp-Stable PersistenceFoundations of Computational Mathematics, 10
A. Cerri, B. Fabio, M. Ferri, Patrizio Frosini, C. Landi (2013)
Betti numbers in multidimensional persistent homology are stable functionsMathematical Methods in the Applied Sciences, 36
T. Ishkhanov (2008)
A topological method for shape comparison2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops
P. Gabriel (1972)
Unzerlegbare Darstellungen Imanuscripta mathematica, 6
G. Carlsson, V. Silva, D. Morozov (2009)
Zigzag persistent homology and real-valued functions
Patrizio Frosini, M. Mulazzani (1999)
Size homotopy groups for computation of natural size distancesBulletin of The Belgian Mathematical Society-simon Stevin, 6
M. Lesnick (2012)
Multidimensional Interleavings and Applications to Topological InferenceArXiv, abs/1206.1365
G. Carlsson, A. Zomorodian (2007)
The Theory of Multidimensional PersistenceDiscrete & Computational Geometry, 42
D. Cohen-Steiner, H. Edelsbrunner, D. Morozov (2006)
Vines and vineyards by updating persistence in linear time
D. Eisenbud (1995)
Commutative Algebra: with a View Toward Algebraic Geometry
c ○ World Scientific Publishing Company AN ALGEBRAIC TOPOLOGICAL METHOD FOR FEATURE IDENTIFICATION
A. Verri, C. Uras, Patrizio Frosini, M. Ferri (1993)
On the use of size functions for shape analysisBiological Cybernetics, 70
A. Blumberg, M. Lesnick (2017)
Universality of the Homotopy Interleaving DistanceArXiv, abs/1705.01690
D. Cohen-Steiner, H. Edelsbrunner, J. Harer (2005)
Stability of Persistence DiagramsDiscrete & Computational Geometry, 37
E Carlsson, G Carlsson, V Silva, S Fortune (2006)
An algebraic topological method for feature identificationInternational Journal of Computational Geometry and Applications, 16
F. Chazal, V. Silva, M. Glisse, S. Oudot (2012)
The Structure and Stability of Persistence ModulesArXiv, abs/1207.3674
(2011)
Persistence-based clustering in riemannian manifolds
G. Carlsson, A. Zomorodian, A. Collins, L. Guibas (2004)
Persistence barcodes for shapesInt. J. Shape Model., 11
M. Lesnick (2011)
The Optimality of the Interleaving Distance on Multidimensional Persistence ModulesArXiv, abs/1106.5305
F. Chazal, D. Cohen-Steiner, Q. Mérigot (2011)
Geometric Inference for Probability MeasuresFoundations of Computational Mathematics, 11
Thomas Judson, Stephen Austin (1968)
Abstract AlgebraThe Mathematical Gazette, 52
Ulrich Bauer, M. Lesnick (2013)
Induced Matchings of Barcodes and the Algebraic Stability of PersistenceProceedings of the thirtieth annual symposium on Computational geometry
Peter Bubenik, V. Silva, Jonathan Scott (2013)
Metrics for Generalized Persistence ModulesFoundations of Computational Mathematics, 15
F. Chazal, D. Cohen-Steiner, L. Guibas, Facundo Mémoli, S. Oudot (2009)
Gromov‐Hausdorff Stable Signatures for Shapes using PersistenceComputer Graphics Forum, 28
W. Crawley-Boevey (2012)
Decomposition of pointwise finite-dimensional persistence modulesarXiv: Representation Theory
H. Locarek-Junge, C. Weihs (2010)
Classification as a Tool for Research
In 2009, Chazal et al. introduced $$\epsilon $$ ϵ -interleavings of persistence modules. $$\epsilon $$ ϵ -interleavings induce a pseudometric $$d_\mathrm{I}$$ d I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $$\epsilon $$ ϵ -interleavings and $$d_\mathrm{I}$$ d I generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view toward applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $$d_\mathrm{I}$$ d I is equal to the bottleneck distance $$d_\mathrm{B}$$ d B . This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $$\epsilon $$ ϵ -interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $$\epsilon $$ ϵ -interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $$d_\mathrm{I}$$ d I satisfies a universality property. This universality result is the central result of the paper. It says that $$d_\mathrm{I}$$ d I satisfies a stability property generalizing one which $$d_\mathrm{B}$$ d B is known to satisfy, and that in addition, if $$d$$ d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $$d\le d_\mathrm{I}$$ d ≤ d I . We also show that a variant of this universality result holds for $$d_\mathrm{B}$$ d B , over arbitrary fields. Finally, we show that $$d_\mathrm{I}$$ d I restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.
Foundations of Computational Mathematics – Springer Journals
Published: Mar 24, 2015
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