Persistence Theory - From Quiver Representations to Data Analysis

* Theoretical foundations: Algebraic persistence* Topological persistence* Stability Applications: Topological inference* Topological inference 2.0* Clustering Signatures for metric spaces* Perspectives: New trends in topological data analysis* Further prospects on the theory* Introduction to quiver theory with a view toward persistence* Bibliography* List of figures* Index

[1]  G. N. Lance,et al.  A General Theory of Classificatory Sorting Strategies: 1. Hierarchical Systems , 1967, Comput. J..

[2]  P. McMullen.  The maximum numbers of faces of a convex polytope , 1970 .

[3]  P. Gabriel.  Unzerlegbare Darstellungen I , 1972 .

[4]  L. Nazarova.  REPRESENTATIONS OF QUIVERS OF INFINITE TYPE , 1973 .

[5]  C. Ringel.  The indecomposable representations of the dihedral 2-groups , 1975 .

[6]  H. Tachikawa,et al.  QF-3 rings. , 1975 .

[7]  Keinosuke Fukunaga,et al.  A Graph-Theoretic Approach to Nonparametric Cluster Analysis , 1976, IEEE Transactions on Computers.

[8]  V. Kac.  Infinite root systems, representations of graphs and invariant theory , 1980 .

[9]  P. Gabriel.  Auslander-Reiten sequences and representation-finite algebras , 1980 .

[10]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[11]  W. B. Johnson,et al.  Extensions of Lipschitz mappings into Hilbert space , 1984 .

[12]  James R. Munkres,et al.  Elements of algebraic topology , 1984 .

[13]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[14]  Cary Webb.  Decomposition of graded modules , 1985 .

[15]  Anil K. Jain,et al.  Algorithms for Clustering Data , 1988 .

[16]  N. Jacobson,et al.  Basic Algebra II , 1989 .

[17]  Ketan Mulmuley,et al.  On levels in arrangements and voronoi diagrams , 1991, Discret. Comput. Geom..

[18]  Emo Welzl,et al.  Smallest enclosing disks (balls and ellipsoids) , 1991, New Results and New Trends in Computer Science.

[19]  Bernd Gärtner.  A Subexponential Algorithm for Abstract Optimization Problems , 1992, FOCS.

[20]  K. Grove.  Critical point theory for distance functions , 1993 .

[21]  Hans-Peter Kriegel,et al.  A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise , 1996, KDD.

[22]  J. Hausmann.  On the Vietoris-Rips complexes and a Cohomology Theory for metric spaces , 1996 .

[23]  Sameer A. Nene,et al.  Columbia Object Image Library (COIL100) , 1996 .

[24]  J. V. van Hateren,et al.  Independent component filters of natural images compared with simple cells in primary visual cortex , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[25]  Yoshua Bengio,et al.  Gradient-based learning applied to document recognition , 1998, Proc. IEEE.

[26]  A. Yakovleva.  Representation of a quiver with relations , 1998 .

[27]  Bernd Gärtner,et al.  Fast and Robust Smallest Enclosing Balls , 1999, ESA.

[28]  M. Gromov.  Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[29]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[30]  J. Latschev.  Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold , 2001 .

[31]  公庄 庸三.  Basic Algebra = 代数学入門 , 2002 .

[32]  Bernard Chazelle,et al.  Shape distributions , 2002, TOGS.

[33]  R. Forman.  A USER'S GUIDE TO DISCRETE MORSE THEORY , 2002 .

[34]  A. Ben Hamza,et al.  Geodesic Object Representation and Recognition , 2003, DGCI.

[35]  C. Villani.  Topics in Optimal Transportation , 2003 .

[36]  Ron Kimmel,et al.  On Bending Invariant Signatures for Surfaces , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[37]  Bernd Gärtner,et al.  Fast Smallest-Enclosing-Ball Computation in High Dimensions , 2003, ESA.

[38]  Joseph S. B. Mitchell,et al.  Approximate minimum enclosing balls in high dimensions using core-sets , 2003, ACM J. Exp. Algorithmics.

[39]  Hans-Peter Kriegel,et al.  Density-Based Clustering in Spatial Databases: The Algorithm GDBSCAN and Its Applications , 1998, Data Mining and Knowledge Discovery.

[40]  Thomas A. Funkhouser,et al.  The Princeton Shape Benchmark , 2004, Proceedings Shape Modeling Applications, 2004..

[41]  D. Ruppert.  The Elements of Statistical Learning: Data Mining, Inference, and Prediction , 2004 .

[42]  Jovan Popovic,et al.  Deformation transfer for triangle meshes , 2004, ACM Trans. Graph..

[43]  André Lieutier,et al.  Any open bounded subset of Rn has the same homotopy type as its medial axis , 2004, Comput. Aided Des..

[44]  Kim Steenstrup Pedersen,et al.  The Nonlinear Statistics of High-Contrast Patches in Natural Images , 2003, International Journal of Computer Vision.

[45]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[46]  D. Theobald.  short communications Acta Crystallographica Section A Foundations of , 2005 .

[47]  Niklas Peinecke,et al.  Laplace-spectra as fingerprints for shape matching , 2005, SPM '05.

[48]  Abubakr Muhammad,et al.  Coverage and hole-detection in sensor networks via homology , 2005, IPSN 2005. Fourth International Symposium on Information Processing in Sensor Networks, 2005..

[49]  Sariel Har-Peled,et al.  Fast construction of nets in low dimensional metrics, and their applications , 2004, SCG.

[50]  Gary L. Miller,et al.  Sparse Voronoi Refinement , 2006, IMR.

[51]  W. Huisinga,et al.  Metastability and Dominant Eigenvalues of Transfer Operators , 2006 .

[52]  Marc E. Pfetsch,et al.  Computing Optimal Morse Matchings , 2006, SIAM J. Discret. Math..

[53]  Daniel Cremers,et al.  Integral Invariants for Shape Matching , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[54]  A. O. Houcine.  On hyperbolic groups , 2006 .

[55]  A. Petrunin.  Semiconcave Functions in Alexandrov???s Geometry , 2013, 1304.0292.

[56]  Leonidas J. Guibas,et al.  BIOINFORMATICS ORIGINAL PAPER doi:10.1093/bioinformatics/btm250 Structural bioinformatics Persistent voids: a new structural metric for membrane fusion , 2022 .

[57]  W. M. Wood-Vasey,et al.  SDSS-III: MASSIVE SPECTROSCOPIC SURVEYS OF THE DISTANT UNIVERSE, THE MILKY WAY, AND EXTRA-SOLAR PLANETARY SYSTEMS , 2011, 1101.1529.

[58]  Raif M. Rustamov,et al.  Laplace-Beltrami eigenfunctions for deformation invariant shape representation , 2007 .

[59]  Afra Zomorodian,et al.  Localized Homology , 2007, IEEE International Conference on Shape Modeling and Applications 2007 (SMI '07).

[60]  Leonidas J. Guibas,et al.  Reconstruction Using Witness Complexes , 2007, SODA '07.

[61]  P. Massart,et al.  Concentration inequalities and model selection , 2007 .

[62]  Frédo Durand,et al.  A Topological Approach to Hierarchical Segmentation using Mean Shift , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[63]  R. Ghrist.  Barcodes: The persistent topology of data , 2007 .

[64]  Takeo Kanade,et al.  Mode-seeking by Medoidshifts , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[65]  Facundo Mémoli,et al.  Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition , 2007, PBG@Eurographics.

[66]  Ulrike von Luxburg,et al.  A tutorial on spectral clustering , 2007, Stat. Comput..

[67]  D. Ringach,et al.  Topological analysis of population activity in visual cortex. , 2008, Journal of vision.

[68]  H. Krause.  Representations of quivers via reflection functors , 2008, 0804.1428.

[69]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[70]  Gary L. Miller,et al.  Linear-Size Meshes , 2008, CCCG.

[71]  Stefano Soatto,et al.  Quick Shift and Kernel Methods for Mode Seeking , 2008, ECCV.

[72]  H. Edelsbrunner,et al.  Homological illusions of persistence and stability , 2008 .

[73]  Marian Mrozek,et al.  Coreduction Homology Algorithm , 2009, Discret. Comput. Geom..

[74]  Eric O. Postma,et al.  Dimensionality Reduction: A Comparative Review , 2008 .

[75]  Gary L. Miller,et al.  Size complexity of volume meshes vs. surface meshes , 2009, SODA.

[76]  David Cohen-Steiner,et al.  Extending Persistence Using Poincaré and Lefschetz Duality , 2009, Found. Comput. Math..

[77]  Danijela Horak,et al.  Persistent homology of complex networks , 2008, 0811.2203.

[78]  Steve Oudot,et al.  Topological inference via meshing , 2010, SoCG '10.

[79]  C. Pichon,et al.  The persistent cosmic web and its filamentary structure II: Illustrations , 2010, 1009.4014.

[80]  E. Coutsias,et al.  Topology of cyclo-octane energy landscape. , 2010, The Journal of chemical physics.

[81]  T. Sousbie.  The persistent cosmic web and its filamentary structure I: Theory and implementation , 2010, 1009.4015.

[82]  Jennifer Gamble,et al.  Exploring uses of persistent homology for statistical analysis of landmark-based shape data , 2010, J. Multivar. Anal..

[83]  Afra Zomorodian,et al.  The tidy set: a minimal simplicial set for computing homology of clique complexes , 2010, SCG.

[84]  S. Mukherjee,et al.  Probability measures on the space of persistence diagrams , 2011 .

[85]  G. Carlsson,et al.  Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival , 2011, Proceedings of the National Academy of Sciences.

[86]  Martin Hilbert,et al.  The World’s Technological Capacity to Store, Communicate, and Compute Information , 2011, Science.

[87]  Primoz Skraba,et al.  Zigzag persistent homology in matrix multiplication time , 2011, SoCG '11.

[88]  Leonidas J. Guibas,et al.  Witnessed k-Distance , 2011, Discrete & Computational Geometry.

[89]  M. Liebeck,et al.  Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras , 2012 .

[90]  Jianzhong Wang,et al.  Geometric Structure of High-Dimensional Data and Dimensionality Reduction , 2012 .

[91]  Don Sheehy,et al.  Linear-Size Approximations to the Vietoris–Rips Filtration , 2012, Discrete & Computational Geometry.

[92]  G. Heo,et al.  Topological Analysis of Variance and the Maxillary Complex , 2012 .

[93]  Quentin Mérigot.  Lower bounds for k-distance approximation , 2013, SoCG '13.

[94]  H. Ombao,et al.  Persistence Landscape of Functional Signal and Its Application to Epileptic Electroencaphalogram Data , 2013 .

[95]  F. Oggier,et al.  An Introduction to Central Simple Algebras and Their Applications to Wireless Communication , 2013 .

[96]  Francesco Vaccarino,et al.  Topological Strata of Weighted Complex Networks , 2013, PloS one.

[97]  Steve Oudot,et al.  Zigzag zoology: rips zigzags for homology inference , 2013, SoCG '13.

[98]  M. Kramár,et al.  Persistence of force networks in compressed granular media. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[99]  Kevin J. Emmett,et al.  Characterizing Scales of Genetic Recombination and Antibiotic Resistance in Pathogenic Bacteria Using Topological Data Analysis , 2014, Brain Informatics and Health.

[100]  Jure Leskovec,et al.  {SNAP Datasets}: {Stanford} Large Network Dataset Collection , 2014 .

[101]  Robert Ghrist,et al.  Elementary Applied Topology , 2014 .

[102]  Sayan Mukherjee,et al.  Fréchet Means for Distributions of Persistence Diagrams , 2012, Discrete & Computational Geometry.

[103]  Don Sheehy,et al.  The Persistent Homology of Distance Functions under Random Projection , 2014, SoCG.

[104]  Ulrich Bauer,et al.  A stable multi-scale kernel for topological machine learning , 2014, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[105]  M. Gameiro,et al.  A topological measurement of protein compressibility , 2014, Japan Journal of Industrial and Applied Mathematics.

[106]  Michael Lesnick,et al.  The Theory of the Interleaving Distance on Multidimensional Persistence Modules , 2011, Found. Comput. Math..

[107]  Steve Oudot,et al.  Zigzag Persistence via Reflections and Transpositions , 2015, SODA.

[108]  Mark W. Johnson,et al.  A Foundation for Props, Algebras, and Modules , 2015 .

[109]  Emerson G. Escolar,et al.  Persistence Modules on Commutative Ladders of Finite Type , 2014, Discret. Comput. Geom..

[110]  Yi Zheng,et al.  Relation between total variation and persistence distance and its application in signal processing , 2016, Adv. Comput. Math..

[111]  R. Ho.  Algebraic Topology , 2022 .