Topology and data

Bulletin of the American Mathematical Society

Volumn 46, Issue 2, 2009, Pages 255-308

  Carlsson, Gunnar ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 70049091784     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-09-01249-X     Document Type: Article

References (65)

1. H. Abdi, Metric multidimensional scaling, in Encyclopedia of Measurement and Statistics, Sage, Thousand Oaks, CA, (2007), pp. 598-605.
2. H. Adams and G. Carlsson, On the non-linear statistics of range image patches,preprint, (2007), available at http://comptop.stanford.edu/preprints/.
3. A. Baddeley, Spatial point processes and their applications, appears in A. Baddeley, I. Bárány, R. Schneider, and W. Weil, editors, Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13-18, 2004,Lecture Notes in Mathematics 1892, Springer-Verlag, Berlin, 2007. ISBN 3-540-38174-0, pp. 1-75. MR2327290 (2008c:60045).
4. S. Ben-David, U. von Luxburg, D. Pál, A sober look on clustering stability,inG.Lugosiand H. Simon, editors, Proceedings of the 19th Annual Conference on Learning Theory (COLT), pages 5-19, Springer-Verlag, Berlin, 2006. MR2277915 (2007i:68102).
5. A. Björner, Topological methods,appearsinHandbook of Combinatorics, Vols. 1, 2, 1819-1872, Elsevier, Amsterdam, 1995. MR1373690 (96m:52012).
6. 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.
7. E. Carlsson, G. Carlsson, and V. de Silva, An algebraic topological method for feature identification, International Journal of Computational Geometry and Applications, 16 (2006), no. 4, pp. 291-314. MR2250511 (2007c:52015).
8. G. Carlsson and V. de Silva, Topological estimation using witness complexes,Symposiumon Point-BasedGraphics,ETH,Zürich, Switzerland, June 2-4, 2004.
9. G. Carlsson and F. Memoli, Persistent Clustering and a Theorem of J. Kleinberg ,Preprint, March 2008.
10. G. Carlsson, A. Zomorodian, A. Collins, and L. Guibas, Persistence barcodes for shapes, International Journal of Shape Modeling, 11 (2005), pp. 149-187.
11. 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.
12. G. Carlsson and A. Zomorodian, The theory of multidimensional persistence,23rdACM Symposium on Computational Geometry, Gyeongju, South Korea, June 6-7, 2007.
13. G. Carlsson and T. Ishkanov, Local structure of spaces of natural images, preprint, (2007), available at http://comptop.stanford.edu/preprints/.
14. G. Carlsson, G. Singh, and A. Zomorodian, Computing multidimensional persistence,in preparation.
15. D. Cohen-Steiner, H. Edelsbrunner and J. Harer, Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103-120. MR2279866 (2008i:68130).
16. D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko, Lipschitz functions have Lp-stable persistence, Found. Comput. Math., to appear.
17. 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.
18. D. Cox, J. Little, and Donal O'Shea, Using Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag, 1998, xii + 499 pages, ISBN 0-387-98492-5. MR1639811 (99h:13033).
19. E. Curtis, Simplicial homotopy theory, Advances in Math. 6 (1971), 107-209. MR0279808 (43:5529)
20. D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes,two volumes, Second Edition, Springer-Verlag, 2003, ISBN 0-387-95541-0. MR950166 (90e:60060).
21. B. Delaunay, Sur la sphere vide, Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk, 7:793-800 1934.
22. J.G. Dumas, F. Heckenbach, B.D. Saunders, and V. Welker, Computing simplicial homology based on efficient Smith normal form algorithms, In Algebra, Geometry, and Software Systems (2003), 177-207. MR2011758 (2004i:55009).
23. D. Dummit and R. Foote, Abstract Algebra. Third edition, John Wiley & Sons, Inc., Hoboken, NJ, 2004. xii+932 pp. ISBN: 0-471-43334-9 00-01 (16-01 20-01). MR2286236 (2007h:00003).
24. H. Edelsbrunner, D. Letscher, and A. Zomorodian, Topological persistence and simplification, Discrete and Computational Geometry 28, 2002, 511-533. MR1949898 (2003m:52019).
25. H. Edelsbrunner and N.R. Shah, Triangulating topological spaces, Tenth Annual ACM Symposium on Computational Geometry (Stony Brook, NY, 1994). Internat. J. Comput. Geom. Appl. 7 (1997), no. 4, 365-378. MR1460843 (98f:57038).
26. B. Efron, Bootstrap methods: Another look at the jack knife, Ann. Statist. 7 (1979), no. 1, pp. 1-26. MR515681 (80b:62021).
27. S. Eilenberg, Singular homology theory, Ann. of Math. (2)45 (1944), 407-447. MR0010970 (6:96f)
28. 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.
29. P. Gabriel and A. Roiter, Representations of Finite-Dimensional Algebras. Translated from the Russian. With a chapter by B. Keller. Reprint of the 1992 English translation. Springer-Verlag, Berlin, 1997. iv+177 pp. ISBN: 3-540-62990-4. MR1475926 (98e:16014).
30. P.G. Goerss and J.F. Jardine, Simplicial homotopy theory, Progress in Mathematics, 174. Birkhäuser Verlag, Basel, 1999. xvi+510 pp. ISBN: 3-7643-6064-X. MR1711612 (2001d:55012).
31. J. Hartigan, Clustering Algorithms, Wiley, New York. MR0405726 (53:9518)
32. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning, Springer, New York, 2001, ISBN: 0-387-95284-5. MR1851606 (2002k:62048)
33. A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN: 0-521-79160-X; 0-521-79540-0. MR1867354 (2002k:55001).
34. 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.
35. 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.
36. D. Hubel, Eye, Brain, and Vision, Scientific American Library, W. H. Freeman, New York, 1995, viii+242pp. ISBN: 0-716-76009-6.
37. P.J. Huber, Projection pursuit, Ann. Statistics (13), 2, (1985), pp. 435-525, with discussion. MR790553 (88b:62118).
38. T. Kenet, D. Bibitchkov, M. Tsodyks, A. Grinvald, and A. Arieli, Spontaneously emerging cortical representations of visual attributes, Nature 425 (2003), 954-956.
39. J.M. Kleinberg, An impossibility theorem for clustering, NIPS 2002: 446-453.
40. 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.
41. 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.
42. R.Y. Liu, J.M. Parelius and K. Singh, Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion and a rejoinder by Liu and Singh), Ann. Statist. Volume 27, Number 3 (1999), 783-858. MR1724033 (2001m:62071).
43. U. von Luxburg, M. Belkin, and O. Bousquet, Consistency of spectral clustering, Annals of Statistics, 36 (2), 555-586, 2008. MR2396807.
44. S. Mac Lane, Categories for the Working Mathematician, Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998. xii+314 pp. ISBN: 0-387-98403-8. MR1712872 (2001j:18001).
45. J.P. May, Simplicial objects in algebraic topology, Reprint of the 1967 original. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1992. viii+161 pp. ISBN: 0-226-51181-2. MR1206474 (93m:55025).
46. P. McCullagh, What is a statistical model? With comments and a rejoinder by the author. Ann. Statist. 30 (2002), no. 5, 1225-1310. MR1936320 (2003j:62006).
47. E. Miller, and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227. Springer-Verlag, New York, 2005. xiv+417 pp. ISBN: 0-387-22356-8. MR2110098 (2006d:13001).
48. R. Miller, Discussion of Projection Pursuit, Ann. Statistics 13,2, 1985, pp. 510-513. (with discussion). MR790553 (88b:62118)
49. J. Milnor, Morse Theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963, vi+153 pp. MR0163331 (29:634).
50. D. Mumford, The dawning of the age of stochasticity,appearsinMathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, 197-218. MR1754778 (2001e:01004).
51. J. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. xvi+413 pp. MR0464128 (57:4063).
52. P. Niyogi, S. Smale, and S. Weinberger, Finding the homology of submanifolds with high confidence from random samples, Discrete and Computational Geometry, vol. 39, nos. 1-3, (2008). MR2383768.
53. 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.
54. M. Penrose, Random Geometric Graphs, Oxford Studies in Probability, 5. Oxford University Press, Oxford, 2003. xiv+330 pp. ISBN: 0-19-850626-0. MR1986198 (2005j:60003).
55. G. Reeb, Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique, C.R. Acad. Sci. Paris 222 (1946), pp. 847-849. MR0015613 (7:446d).
56. S.T. Roweis and L.K. Saul, Nonlinear dimensionality reduction by locally linear embedding, Science 290 (2000) (December), pp. 2323-2326.
57. V. de Silva, R. Ghrist, Coverage in sensor networks via persistent homology, Algebraic and Geometric Topology, 7, 2007, pp. 339-358. MR2308949 (2008c:55008).
58. B.W. Silverman, Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability. Chapman & Hall, London, 1986. x+175 pp. ISBN: 0-412-24620-1. MR848134 (87k:62074).
59. G. 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.
60. G. 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.
61. J.B. Tenenbaum, V. de Silva and J.C. Langford, A global geometric framework for nonlinear dimensionality reduction, Science 290 (2000) (December), pp. 2319-2323.
62. 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.
63. B. Wandell, Foundations of Vision, Sinauer Associates, Sunderland, Mass., 1995, xvi+476pp., ISBN:0-878-93853-2.
64. A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete and Computational Geometry, 33 (2), 2005, pp. 247-274. MR2121296 (2005j:55004).
65. A. Zomorodian and G. Carlsson, Localized homology, Computational Geometry: Theory and Applications, 41,(3), pp. 126-148, 2008. MR2442490.