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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Topology and data

Author: Gunnar Carlsson
Journal: Bull. Amer. Math. Soc. 46 (2009), 255-308
Published electronically: January 29, 2009
MathSciNet review: 2476414
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    abdi H. Abdi, Metric multidimensional scaling, in Encyclopedia of Measurement and Statistics, Sage, Thousand Oaks, CA, (2007), pp. 598-605. range H. Adams and G. Carlsson, On the non-linear statistics of range image patches, preprint, (2007), available at
  • Adrian Baddeley, Spatial point processes and their applications, Stochastic geometry, Lecture Notes in Math., vol. 1892, Springer, Berlin, 2007, pp. 1–75. MR 2327290, DOI
  • Shai Ben-David, Ulrike von Luxburg, and Dávid Pál, A sober look at clustering stability, Learning theory, Lecture Notes in Comput. Sci., vol. 4005, Springer, Berlin, 2006, pp. 5–19. MR 2277915, DOI
  • A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690
  • bowman G. R. Bowman, X. Huang, Y. Yao, J. Sun, G. Carlsson, L. J. Guibas, and V. S. Pande, Structural insight into RNA hairpin folding intermediates, Journal of the American Chemical Society Communications, July, 2008.
  • Erik Carlsson, Gunnar Carlsson, and Vin de Silva, An algebraic topological method for feature identification, Internat. J. Comput. Geom. Appl. 16 (2006), no. 4, 291–314. MR 2250511, DOI
  • witness G. Carlsson and V. de Silva, Topological estimation using witness complexes, Symposium on Point-Based Graphics, ETH, Zürich, Switzerland, June 2-4, 2004. memolicarlsson G. Carlsson and F. Memoli, Persistent Clustering and a Theorem of J. Kleinberg , Preprint, March 2008. shapes G. Carlsson, A. Zomorodian, A. Collins, and L. Guibas, Persistence barcodes for shapes, International Journal of Shape Modeling, 11 (2005), pp. 149-187. patches G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, On the local behavior of spaces of natural images, International Journal of Computer Vision, (76), 1, 2008, pp. 1-12. multid G. Carlsson and A. Zomorodian, The theory of multidimensional persistence, 23rd ACM Symposium on Computational Geometry, Gyeongju, South Korea, June 6-7, 2007. tigran G. Carlsson and T. Ishkanov, Local structure of spaces of natural images, preprint, (2007), available at grobmultid G. Carlsson, G. Singh, and A. Zomorodian, Computing multidimensional persistence, in preparation.
  • David Cohen-Steiner, Herbert Edelsbrunner, and John Harer, Stability of persistence diagrams, Discrete Comput. Geom. 37 (2007), no. 1, 103–120. MR 2279866, DOI
  • stabilitytwo D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko, Lipschitz functions have $L_p$-stable persistence, Found. Comput. Math., to appear. curves A. Collins, A. Zomorodian, G. Carlsson, and L. Guibas, A barcode shape descriptor for curve point cloud data, Computers and Graphics, Volume 28, 2004, pp. 881–894.
  • David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811
  • Edward B. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107–209 (1971). MR 279808, DOI
  • D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Series in Statistics, Springer-Verlag, New York, 1988. MR 950166
  • delaunay B. Delaunay, Sur la sphere vide, Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793-800 1934.
  • Jean-Guillaume Dumas, Frank Heckenbach, David Saunders, and Volkmar Welker, Computing simplicial homology based on efficient Smith normal form algorithms, Algebra, geometry, and software systems, Springer, Berlin, 2003, pp. 177–206. MR 2011758
  • David S. Dummit and Richard M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR 2286236
  • Herbert Edelsbrunner, David Letscher, and Afra Zomorodian, Topological persistence and simplification, Discrete Comput. Geom. 28 (2002), no. 4, 511–533. Discrete and computational geometry and graph drawing (Columbia, SC, 2001). MR 1949898, DOI
  • Herbert Edelsbrunner and Nimish R. Shah, Triangulating topological spaces, Internat. J. Comput. Geom. Appl. 7 (1997), no. 4, 365–378. Tenth Annual ACM Symposium on Computational Geometry (Stony Brook, NY, 1994). MR 1460843, DOI
  • B. Efron, Bootstrap methods: another look at the jackknife, Ann. Statist. 7 (1979), no. 1, 1–26. MR 515681
  • Samuel Eilenberg, Singular homology theory, Ann. of Math. (2) 45 (1944), 407–447. MR 10970, DOI
  • frosini P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognition and Image Analysis, vol. 9 (4) (1999), pp. 596-603.
  • P. Gabriel and A. V. Roiter, Representations of finite-dimensional algebras, Springer-Verlag, Berlin, 1997. Translated from the Russian; With a chapter by B. Keller; Reprint of the 1992 English translation. MR 1475926
  • Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Progress in Mathematics, vol. 174, Birkhäuser Verlag, Basel, 1999. MR 1711612
  • John A. Hartigan, Clustering algorithms, John Wiley & Sons, New York-London-Sydney, 1975. Wiley Series in Probability and Mathematical Statistics. MR 0405726
  • Trevor Hastie, Robert Tibshirani, and Jerome Friedman, The elements of statistical learning, Springer Series in Statistics, Springer-Verlag, New York, 2001. Data mining, inference, and prediction. MR 1851606
  • Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
  • vanhateren J. H. van Hateren and A. van der Schaaf, Independent component filters of natural images compared with simple cells in primary visual cortex, Proc. R. Soc. Lond., vol. B 265 (1998), 359-366. docking J. Headd, Y.-H. A. Ban, H. Edelsbrunner, M. Vaidya and J. Rudolph. Protein-protein interfaces: Properties, preferences, and projections, Protein Research, to appear, 2007. hubel D. Hubel, Eye, Brain, and Vision, Scientific American Library, W. H. Freeman, New York, 1995, viii+242pp. ISBN: 0-716-76009-6.
  • Peter J. Huber, Projection pursuit, Ann. Statist. 13 (1985), no. 2, 435–525. With discussion. MR 790553, DOI
  • kenet T. Kenet, D. Bibitchkov, M. Tsodyks, A. Grinvald, and A. Arieli, Spontaneously emerging cortical representations of visual attributes, Nature 425 (2003), 954-956. kleinbergJ.M. Kleinberg, An impossibility theorem for clustering, NIPS 2002: 446-453. laplacians S. Lafon and A.B. Lee, Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning, and data set parametrization, IEEE Transactions on Pattern Analysis and Machine Intelligence 28, 9 (2006), pp. 1393-1403. lee A.B. Lee, K.S. Pedersen, and D. Mumford, The nonlinear statistics of high-contrast patches in natural images, International Journal of Computer Vision (54), No. 1-3, August 2003, pp. 83-103.
  • Regina Y. Liu, Jesse M. Parelius, and Kesar Singh, Multivariate analysis by data depth: descriptive statistics, graphics and inference, Ann. Statist. 27 (1999), no. 3, 783–858. With discussion and a rejoinder by Liu and Singh. MR 1724033, DOI
  • Ulrike von Luxburg, Mikhail Belkin, and Olivier Bousquet, Consistency of spectral clustering, Ann. Statist. 36 (2008), no. 2, 555–586. MR 2396807, DOI
  • Saunders Mac Lane, Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998. MR 1712872
  • J. Peter May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. Reprint of the 1967 original. MR 1206474
  • Peter McCullagh, What is a statistical model?, Ann. Statist. 30 (2002), no. 5, 1225–1310. With comments and a rejoinder by the author. MR 1936320, DOI
  • Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
  • Peter J. Huber, Projection pursuit, Ann. Statist. 13 (1985), no. 2, 435–525. With discussion. MR 790553, DOI
  • J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
  • David Mumford, The dawning of the age of stochasticity, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 197–218. MR 1754778
  • James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
  • Partha Niyogi, Stephen Smale, and Shmuel Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete Comput. Geom. 39 (2008), no. 1-3, 419–441. MR 2383768, DOI
  • clique G. Palla, I. Derènyi, I. Farkas, and T. Vicsek, Uncovering the overlapping community structure of complex networks in nature and society, Nature, Volume 435, 9 June 2005, pp. 814-818.
  • Mathew Penrose, Random geometric graphs, Oxford Studies in Probability, vol. 5, Oxford University Press, Oxford, 2003. MR 1986198
  • Georges Reeb, Sur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numérique, C. R. Acad. Sci. Paris 222 (1946), 847–849 (French). MR 15613
  • lleS.T. Roweis and L.K. Saul, Nonlinear dimensionality reduction by locally linear embedding, Science 290 (2000) (December), pp. 2323-2326.
  • Vin de Silva and Robert Ghrist, Coverage in sensor networks via persistent homology, Algebr. Geom. Topol. 7 (2007), 339–358. MR 2308949, DOI
  • B. W. Silverman, Density estimation for statistics and data analysis, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1986. MR 848134
  • visionG. Singh, F. Memoli, T. Ishkhanov, G. Carlsson, G. Sapiro and D. Ringach, Topological Structure of Population Activity in Primary Visual Cortex, Journal of Vision, Volume 8, Number 8, Article 11, pp. 1-18, 2008. mapperG. Singh, F. Memoli and G. Carlsson, Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition, Point Based Graphics 2007, Prague, September 2007. isomapJ.B. Tenenbaum, V. de Silva and J.C. Langford, A global geometric framework for nonlinear dimensionality reduction, Science 290 (2000) (December), pp. 2319-2323. tsodyks M. Tsodyks, T. Kenet, A. Grinvald, and A. Arieli, Linking spontaneous activity of single cortical neurons and the underlying functional architecture, Science 286, (1999), pp. 1943-1996. wandell B. Wandell, Foundations of Vision, Sinauer Associates, Sunderland, Mass., 1995, xvi+476pp., ISBN:0-878-93853-2.
  • Afra Zomorodian and Gunnar Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005), no. 2, 249–274. MR 2121296, DOI
  • Afra Zomorodian and Gunnar Carlsson, Localized homology, Comput. Geom. 41 (2008), no. 3, 126–148. MR 2442490, DOI

Additional Information

Gunnar Carlsson
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
MR Author ID: 45435

Received by editor(s): August 1, 2008
Published electronically: January 29, 2009
Additional Notes: Research supported in part by DARPA HR 0011-05-1-0007 and NSF DMS 0354543
Article copyright: © Copyright 2009 American Mathematical Society