Diffusion wavelets

Applied and Computational Harmonic Analysis

Volumn 21, Issue 1, 2006, Pages 53-94

  Coifman, Ronald R ^a   Maggioni, Mauro ^a  

^a YALE UNIVERSITY   (United States)

Author keywords

Diffusion semigroups; Fast Multipole Method; Laplace Beltrami operator; Matrix compression; Multiresolution; Multiscale analysis; Spectral graph theory; Wavelets; Wavelets on graphs; Wavelets on manifolds

Indexed keywords

ALGORITHMS; DATA REDUCTION; DIFFUSION; DIGITAL STORAGE; FOURIER TRANSFORMS; GRAPH THEORY; GREEN'S FUNCTION; INSTRUMENT SCALES; NUMERICAL METHODS; SIGNAL PROCESSING; SPECTRUM ANALYSIS;

DIFFUSION SEMIGROUPS; FAST MULTIPOLE METHOD; LAPLACE-BELTRAMI OPERATOR; MATRIX COMPRESSION; MULTIRESOLUTION; MULTISCALE ANALYSIS; SPECTRAL GRAPH THEORY; WAVELETS; WAVELETS ON GRAPHS; WAVELETS ON MANIFOLDS;

WAVELET TRANSFORMS;

EID: 33745332989     PISSN: 10635203     EISSN: 1096603X     Source Type: Journal    
DOI: 10.1016/j.acha.2006.04.004     Document Type: Article

References (94)

1. Greengard L., and Rokhlin V. A fast algorithm for particle simulations. J. Comput. Phys. 73 (1987) 325-348
2. Beylkin G., Coifman R.R., and Rokhlin V. Fast wavelet transforms and numerical algorithms. Comm. Pure Appl. Math. 44 (1991) 141-183
3. S. Lafon, Diffusion maps and geometric harmonics, Ph.D. thesis, Yale University, 2004
4. R. Coifman, S. Lafon, Geometric harmonics, Technical report, Yale University, 2003
5. Stein E. Topics in Harmonic Analysis Related to the Littlewood-Paley Theory (1970), Princeton Univ. Press
6. Coifman R.R., and Meyer Y. Wavelets: Calderón-Zygmund and Multilinear Operators (1993), Cambridge Univ. Press
7. David G., and Journé J. A boundedness criterion for generalized Calderón-Zygmund operators. Ann. of Math 120 2 (1984) 371-391
8. David G., Journé J., and Semmes S. Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Mat. Iberoamericana 1 (1985) 1-56
9. Gilbert J., Han Y., Hogan J., Lakey J., Weiland D., and Weiss G. Smooth molecular decompositions of functions and singular integral operators. Mem. Amer. Math. Soc. 156 (2002)
10. Han Y. Calderón-type reproducing formula and the Tb theorem. Rev. Mat. Iberoamericana 10 1 (1994) 51-91
11. Nahmod A. Generalized uncertainty principles on spaces of homogeneous type. J. Funct. Anal. 119 (1994) 171-209
12. J.C. Bremer, R.R. Coifman, M. Maggioni, A.D. Szlam, Diffusion wavelet packets, Technical report YALE/DCS/TR-1304, Yale University
13. Chung F. Spectral Graph Theory, vol. 92 (1997), CBMS, Amer. Math. Soc.
14. Davies E. Heat Kernels and Spectral Theory (1989), Cambridge Univ. Press
15. Li P., and Yau S.T. On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986) 153-201
16. C.E. Priebe, D.J. Marchette, Y. Park, E.J. Wegman, J.L. Solka, D.A. Socolinsky, D. Karakos, K.W. Church, R. Guglielmi, R.R. Coifman, D. Link, D.M. Healy, M.Q. Jacobs, A. Tsao, Iterative denoising for cross-corpus discovery, 2004, in preparation
17. R.R. Coifman, M. Maggioni, Multiscale analysis of document corpora, in preparation
18. Belkin M., and Niyogi P. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 6 15 (2003) 1373-1396
19. M. Brand, Continuous nonlinear dimensionality reduction by kernel eigenmaps, in: Proc. IJCAI, 2003
20. R. Coifman, S. Lafon, Diffusion maps, Appl. Comput. Harmon. Anal., in press
21. Coifman R.R., and Weiss G.L. Analyse harmonique noncommutative sur certains espaces homogenes (1971), Springer-Verlag
22. Macias R., and Segovia C. Lipschitz functions on spaces of homogeneous type. Adv. Math. 33 (1979) 257-270
23. Macias R., and Segovia C. A decomposition into atoms of distribution on spaces of homogeneous type. Adv. Math. 33 (1979) 271-309
24. Bownik M. Anisotropic Hardy spaces and wavelets. Mem. Amer. Math. Soc. 781 (2003)
25. Nagel A., Stein E., and Wainger S. Balls and metrics defined by vector fields I: Basic properties. Acta Math. 155 (1985) 103-147
26. Christ M. A T (b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 2 60/61 (1990) 601-628
27. David G. Wavelets and Singular Integrals on Curves and Surfaces (1991), Springer-Verlag
28. Shi J., and Malik J. Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22 8 (2000) 888-905
29. W. Goetzmann, P.W. Jones, M. Maggioni, J. Walden, Beauty is in the eye of the beholder, preprint
30. Laugesen R., Wilson E., Weaver N., and Weiss G. A characterization of the higher dimensional groups associated with continuous wavelets. J. Geom. Anal. 12 1 (2002) 89-102
31. Maggioni M. Wavelet frames on groups and hypergroups via discretization of Calderón formulas. Monatsh. Math. 143 (2004) 299-331
32. Trimeche K. Generalized Wavelets and Hypergroups (1997), Taylor & Francis
33. M. Belkin, P. Niyogi, Using manifold structure for partially labelled classification, in: Adv. NIPS, vol. 15, 2003
34. P. Niyogi, X. He, Locality preservity projections, Technical report TR-2002-09, University of Chicago, 2002
35. Donoho D.L., and Grimes C. Hessian eigenmaps: New locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Sci. (2003) 5591-5596 Also technical report, Stanford University
36. J. Ham, D. Lee, S. Mika, B. Schölkpf, A kernel view of the dimensionality reduction of manifolds, Technical report, Max-Planck-Institut für Biologische Kybernetik, Tübingen, July 2003
37. Tenenbaum J., de Silva V., and Langford J. A global geometric framework for nonlinear dimensionality reduction. Science 290 (2000) 2319-2323
38. Y. Bengio, J. Paiement, P. Vincent, Out-of-sample extensions of lle, isomap, mds, eigenmaps, and spectral clustering, in: NIPS, vol. 16, 2004
39. Riemenschneider S., and Shen Z. Box Splines, Cardinal Series, and Wavelets (1991), Academic Press
40. Linial N., London E., and Rabinovich Y. The geometry of graphs and some of its algorithmic applications. Combinatorica 15 (1995) 215-245
41. A. Nahmod, Geometry of operators and spectral analysis, Ph.D. thesis, Yale University, 1991
42. H. Cheng, Z. Gimbutas, P. Martinsson, V. Rokhlin, On the compression of low rank matrices, Technical report YALEU/DCS/RR-1251, Yale University, July 2003
43. P.G. Martinsson, V. Rokhlin, A fast direct solver for boundary integral equations in two dimensions, Technical report YALEU/DCS/RR-1264, Yale University, 2004
44. Hackbusch W. A sparse matrix arithmetic based on H-matrices I. Introduction to H-matrices. Computing 62 2 (1999) 89-108
45. 2-matrices. Computing 69 1 (2002) 1-35
46. A. Brandt, S. McCormick, J. Ruge, Algebraic multigrid (AMG) for automatic multigrid solution with applications to geodetic computations, Technical report, Institute for Computational Studies, Fort Colins, CO, 1982
47. O. Livne, Multiscale eigenbasis algorithms, Ph.D. thesis, Department Computer Sciences & Applied Mathematics, Weizmann Institute, Rehovot, 2000
48. Heinonen J. Lectures on Analysis on Metric Spaces (2001), Springer-Verlag, New York
49. Mallat S.G. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11 7 (1989) 674-693
50. Meyer Y. Ondelettes et operatéurs (1990), Hermann, Paris
51. Daubechies I. Ten Lectures on Wavelets (1992), SIAM, Philadelphia
52. Coifman R.R. Multiresolution analysis in nonhomogeneous media. In: Combes J.M., Grossman A., and Tchamitchian Ph. (Eds). Wavelets: Time-Frequency Methods and Phase Space (1989), Springer-Verlag, New York
53. Cohen A., Daubechies I., and Faveau J. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. XLV (1992) 485-560
54. M. Maggioni, J.C. Bremer Jr., R.R. Coifman, A.D. Szlam, Biorthogonal diffusion wavelets for multiscale representations on manifolds and graphs, in: Wavelet XI, in: Proc. SPIE, vol. 5914, 2005, 59141M
55. J.C. Bremer, R.R. Coifman, M. Maggioni, Biorthogonal diffusion wavelets, in preparation
56. M. Maggioni, A.D. Szlam, R.R. Coifman, J.C. Bremer Jr., Diffusion-driven multiscale analysis on manifolds and graphs: Top-down and bottom-up constructions, in: Wavelet XI, in: Proc. SPIE, vol. 5914, 2005, 59141D
57. Björck A. Numerics of Gram-Schmidt orthogonalization. Linear Algebra Appl. 197 (1994) 297-316
58. Golub G., and Loan C.V. Matrix Computations (1989), Johns Hopkins Univ. Press
59. Daniel J., Gragg W., Kaufman L., and Stewart G. Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorization. Math. Comput. 30 (1976) 772-795
60. Indyk P., and Motwani R. Approximate nearest neighbors: Towards removing the curse of dimensionality. 1998, pp. 604-613. http://citeseer.ist.psu.edu/article/indyk98approximate.html
61. R. Krauthgamer, J. Lee, M. Mende, A. Naor, Measured descent: A new embedding method for finite metrics, in preparation
62. n as unconditional bases for Hardy spaces, in: Conference in Honor of A. Zygmund of Wadsworth Math. Ser., 1982, p. 475
63. Gu M., and Eisenstat S. Efficient algorithms for computing a strong rank-revealing QR factorization. SIAM J. Sci. Comput. 17 4 (1996) 848-869
64. L. Grasedyck, W. Hackbusch, S.L. Borne, Adaptive geometrically balanced clustering of h-matrices, Computing (2004), in press
65. N. Sidorova, O.G. Smolyanov, H. Weizsäcker, O. Wittich, Brownian motion close to submanifold of Riemannian manifolds, preprint, 2003
66. Smolyanov O.G., v Weizsäcker H., and Wittich O. Brownian motion on a manifold as limit of stepwise conditioned standard Brownian motions. Stochastic Processes, Physics and Geometry: New Interplays. Can. Math. Soc. Conf. Proc. vol. 29 (2000), Amer. Math. Soc. 589-602
67. Daubechies I., Guskov I., Schröder P., and Sweldens W. Wavelets on irregular point sets. Philos. Trans. R. Soc. London A 357 1760 (1999) 2397-2413
68. Simoens J., and Vandewalle S. A stabilized lifting construction of wavelets on irregular meshes on the interval. SIAM J. Sci. Comput. 4 24 (2003) 1356-1378
69. D.L. Donoho, I. Johnstone, Ideal denoising in an orthonormal basis chosen from a library of bases, Technical report, Stanford University, 1994
70. R.R. Coifman, M. Maggioni, Multiscale data analysis with diffusion wavelets, Technical report YALE/DCS/TR-1335, Yale University, September 2005
71. R. Coifman, S. Lafon, Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions, Appl. Comput. Harmon. Anal., in press
72. Slepian D., and Pollack H. Prolate spheroidal wave functions, Fourier analysis and uncertainty I. Bell Syst. Tech. J. 40 (1961) 43-64
73. Slepian D. Prolate spheroidal wave functions, Fourier analysis and uncertainty IV: Extensions to many dimensions; generalized prolate spheroidal wave functions. Bell Syst. Tech. J. 43 (1964) 3009-3058
74. Fowlkes C., Belongie S., Chung F., and Malik J. Spectral grouping using the Nyström method. IEEE Trans. Pattern Anal. Mach. Intell. 26 2 (2004) 214-225
75. C. Fowlkes, S. Belongie, J. Malik, Efficient spatiotemporal grouping using the Nyström method, CVPR
76. Williams C.K.I., and Seeger M. Using the Nyström method to speed up kernel machines. NIPS (2000) 682-688
77. Karni Z., and Gotsman C. Spectral compression of mesh geometry. Proc. 27th Annual Conference on Computer Graphics and Interactive Techniques. SIGGRAPH'00 (2000), ACM Press/Addison-Wesley Publishing Co. 279-286
78. Taubin G. A signal processing approach to fair surface design. Proc. 22nd Annual Conference on Computer Graphics and Interactive Techniques. SIGGRAPH'95 (1995), ACM Press 351-358
79. Taubin G., Zhang T., and Golub G.H. Optimal surface smoothing as filter design. ECCV vol. 1 (1996) 283-292
80. Guskov I., Sweldens W., and Schröder P. Multiresolution signal processing for meshes. Proc. 26th Annual Conference on Computer Graphics and Interactive Techniques. SIGGRAPH'99 (1999), ACM Press/Addison-Wesley Publishing Co 325-335
81. Desbrun M., Meyer M., Schröder P., and Barr A.H. Implicit fairing of irregular meshes using diffusion and curvature flow. Proc. 26th Annual Conference on Computer Graphics and Interactive Techniques. SIGGRAPH'99 (1999), ACM Press/Addison-Wesley Publishing Co. 317-324
82. I. Guskov, W. Sweldens, P. Schröder, Multiresolution signal processing for meshes, in: Computer Graphics Proceedings, SIGGRAPH'99, 1999, pp. 325-334
83. P. Schröder, W. Sweldens, Spherical wavelets: Efficiently representing functions on the sphere, in: Computer Graphics Proceedings, SIGGRAPH'95, 1995, pp. 161-172
84. Sweldens W. The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3 2 (1996) 186-200
85. Sweldens W. The lifting scheme: A construction of second generation wavelets. SIAM J. Math. Anal. 29 2 (1997) 511-546
86. Kovačević J., and Sweldens W. Wavelet families of increasing order in arbitrary dimensions. IEEE Trans. Image Process. 9 3 (2000) 480-496
87. D. Zorin, P. Schröder, W. Sweldens, Interpolating subdivision for meshes with arbitrary topology, in: Computer Graphics Proceedings, SIGGRAPH'96, 1996, pp. 189-192
88. Lounsbery M., DeRose T.D., and Warren J. Multiresolution analysis for surfaces of arbitrary topological type. ACM Trans. Graph. 16 1 (1997) 34-73
89. Lemarie P. Base d'ondelettes sur les groups de Lie stratifies. Bull. Soc. Math. France (117) (1989) 211-232
90. S. Mahadevan, M. Maggioni, Value function approximation with diffusion wavelets and Laplacian eigenfunctions, Technical report TR-2005-38, University of Massachusetts, Department of Computer Science, in: Proc. NIPS, 2005
91. M. Maggioni, S. Mahadevan, Fast direct policy evaluation using multiscale analysis of Markov diffusion processes, Technical report TR-2005-39, University of Massachusetts, Department of Computer Science, accepted at ICML 2006, 2005
92. S. Mahadevan, K. Ferguson, S. Osentoski, M. Maggioni, Simultaneous learning of representation and control in continuous domains, in preparation
93. S. Mahadevan, M. Maggioni, Proto-value functions: A spectral framework for solving Markov decision processes, J. Mach. Learn. Res., submitted for publication
94. M. Maggioni, S. Mahadevan, Multiscale diffusion bases for policy iteration in Markov decision processes, J. Mach. Learn. Res., submitted for publication