Group Invariant Scattering
This paper constructs translation‐invariant operators on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$, which are Lipschitz‐continuous to the action of diffeomorphisms. A scattering propagator is a path‐ordered product of nonlinear and noncommuting operators, each of which c...
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| Published in | Communications on pure and applied mathematics Vol. 65; no. 10; pp. 1331 - 1398 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Hoboken
Wiley Subscription Services, Inc., A Wiley Company
01.10.2012
John Wiley and Sons, Limited |
| Subjects | |
| Online Access | Get full text |
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| Abstract | This paper constructs translation‐invariant operators on
$\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$, which are Lipschitz‐continuous to the action of diffeomorphisms. A scattering propagator is a path‐ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz‐continuous to the action of C2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high‐order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L2(G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$
and on L2(SO(d)) defines a translation‐ and rotation‐invariant scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$. © 2012 Wiley Periodicals, Inc. |
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| AbstractList | This paper constructs translation-invariant operators on L2(Rd), which are Lipschitz-continuous to the action of diffeomorphisms. A scattering propagator is a path-ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz-continuous to the action of C2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high-order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L2(G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L2(Rd) and on L2(SO(d)) defines a translation- and rotation-invariant scattering on L2(Rd). [PUBLICATION ABSTRACT] This paper constructs translation‐invariant operators on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$, which are Lipschitz‐continuous to the action of diffeomorphisms. A scattering propagator is a path‐ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz‐continuous to the action of C2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high‐order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L2(G), where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ and on L2(SO(d)) defines a translation‐ and rotation‐invariant scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$. © 2012 Wiley Periodicals, Inc. This paper constructs translation‐invariant operators on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ , which are Lipschitz‐continuous to the action of diffeomorphisms. A scattering propagator is a path‐ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz‐continuous to the action of C 2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high‐order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L 2 ( G ), where G is a compact Lie group, and are invariant under the action of G . Combining a scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ and on L 2 ( SO ( d )) defines a translation‐ and rotation‐invariant scattering on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ . © 2012 Wiley Periodicals, Inc. |
| Author | Mallat, Stéphane |
| Author_xml | – sequence: 1 givenname: Stéphane surname: Mallat fullname: Mallat, Stéphane email: mallat@cmap.polytechnique.fr organization: Ecole Polytechnique, CMAP, 91128 PALAISEAU CEDEX, FRANCE |
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| CODEN | CPMAMV |
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| Cites_doi | 10.1111/j.1749-6632.1988.tb32998.x 10.1109/ISCAS.2010.5537907 10.1137/050622729 10.1137/S0036141002404838 10.4171/RMI/17 10.1016/S0005-1098(98)00019-3 10.1038/14819 10.1017/CBO9780511609565 10.1109/CVPR.2011.5995635 10.1007/BF02392052 10.1007/s12220-010-9150-3 10.1090/cbms/079 10.1515/9781400881871 |
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| References | Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. (English summary) With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, N.J., 1993. Olver, P. J. Equivalence, invariants and symmetry. Cambridge University Press, Cambridge, 1995. Frankel, T. The geometry of physics: an introduction. 2nd ed. Cambridge University Press, Cambridge, 2004. Meyer, Y. Wavelets and operators. Cambridge Studies in Advanced Mathematics, Vol. 37. Cambridge University Press, Cambridge, 1992. Bruna, J.; Mallat, S. Invariant scattering convolution networks. IEEE Trans. PAMI, to appear. Bruna, J.; Mallat, S. Classification with scattering operators. 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) ( 2011), 1561-1566. doi:10.1109/CVPR.2011.5995635 Frazier, M.; Jawerth, B.; Weiss, G. Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics, 79. Published for the Conference Board of the Mathematical Sciences, Washington, D.C., by the American Mathematical Society, Providence, R.I., 1991. Hörmander, L. Fourier integral operators. I. Acta Math. 127 ( 1971), no. 1, 79-183. Willard, S. General topology. Addison-Wesley, Reading, Mass.-London-Don Mills, Ont., 1970. Stein, E. M. Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, 63. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. Rahman, I. U.; Drori, I.; Stodden, V. C.; Donoho, D. L.; Schrder, P. Multiscale representations for manifold-valued data. Multiscale Model. Simul. 4 ( 2005), no. 4, 1201-1232. Lohmiller, W.; Slotine, J. J. E. On contraction analysis for nonlinear systems. Automatica 34 ( 1998), no. 6, 683-696. Riesenhuber, M.; Poggio, T. Hierarchical models of object recognition in cortex. Nat. Neurosci. 2 ( 1999), 1019-1025. Trouv, A.; Younes, L. Local geometry of deformable templates. SIAM J. Math. Anal. 37 ( 2005), no. 1, 17-59 (electronic). doi:10.1137/S0036141002404838 Bouvrie, J.; Rosasco, L.; Poggio, T. On invariance in hierarchical models. Advances in Neural Information Processing Systems 22, 162-170, 2009. David, G.; Journé, J.-L.; Semmes, S. Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. (French) [Calderón-Zygmund operators, para-accretive functions and interpolation] Rev. Mat. Iberoamericana 1 ( 1985), no. 4, 1-56. Geller, D.; Pesenson, I. Z. Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21 ( 2011), no. 2, 334-371. 1985; 1 2012 2011 2009 1971; 127 2005; 4 2011; 21 1995 2004 1993 1992; 37 1999; 2 1970 1991 2005; 37 1998; 34 Frankel T. (e_1_2_1_7_2) 2004 Bouvrie J. (e_1_2_1_3_2) 2009 Bruna J. (e_1_2_1_5_2) e_1_2_1_6_2 e_1_2_1_4_2 Willard S. (e_1_2_1_23_2) 1970 e_1_2_1_2_2 e_1_2_1_11_2 e_1_2_1_22_2 e_1_2_1_12_2 e_1_2_1_20_2 e_1_2_1_10_2 e_1_2_1_15_2 e_1_2_1_16_2 Stein E. M. (e_1_2_1_21_2) 1993 e_1_2_1_13_2 e_1_2_1_14_2 e_1_2_1_19_2 e_1_2_1_8_2 e_1_2_1_17_2 e_1_2_1_9_2 e_1_2_1_18_2 |
| References_xml | – reference: Frankel, T. The geometry of physics: an introduction. 2nd ed. Cambridge University Press, Cambridge, 2004. – reference: David, G.; Journé, J.-L.; Semmes, S. Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. (French) [Calderón-Zygmund operators, para-accretive functions and interpolation] Rev. Mat. Iberoamericana 1 ( 1985), no. 4, 1-56. – reference: Riesenhuber, M.; Poggio, T. Hierarchical models of object recognition in cortex. Nat. Neurosci. 2 ( 1999), 1019-1025. – reference: Willard, S. General topology. Addison-Wesley, Reading, Mass.-London-Don Mills, Ont., 1970. – reference: Hörmander, L. Fourier integral operators. I. Acta Math. 127 ( 1971), no. 1, 79-183. – reference: Meyer, Y. Wavelets and operators. Cambridge Studies in Advanced Mathematics, Vol. 37. Cambridge University Press, Cambridge, 1992. – reference: Trouv, A.; Younes, L. Local geometry of deformable templates. SIAM J. Math. Anal. 37 ( 2005), no. 1, 17-59 (electronic). doi:10.1137/S0036141002404838 – reference: Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. (English summary) With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, N.J., 1993. – reference: Frazier, M.; Jawerth, B.; Weiss, G. Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics, 79. Published for the Conference Board of the Mathematical Sciences, Washington, D.C., by the American Mathematical Society, Providence, R.I., 1991. – reference: Bruna, J.; Mallat, S. Classification with scattering operators. 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) ( 2011), 1561-1566. doi:10.1109/CVPR.2011.5995635 – reference: Rahman, I. U.; Drori, I.; Stodden, V. C.; Donoho, D. L.; Schrder, P. Multiscale representations for manifold-valued data. Multiscale Model. Simul. 4 ( 2005), no. 4, 1201-1232. – reference: Lohmiller, W.; Slotine, J. J. E. On contraction analysis for nonlinear systems. Automatica 34 ( 1998), no. 6, 683-696. – reference: Olver, P. J. Equivalence, invariants and symmetry. Cambridge University Press, Cambridge, 1995. – reference: Geller, D.; Pesenson, I. Z. Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21 ( 2011), no. 2, 334-371. – reference: Bouvrie, J.; Rosasco, L.; Poggio, T. On invariance in hierarchical models. Advances in Neural Information Processing Systems 22, 162-170, 2009. – reference: Bruna, J.; Mallat, S. Invariant scattering convolution networks. IEEE Trans. PAMI, to appear. – reference: Stein, E. M. Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, 63. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1970. – article-title: Invariant scattering convolution networks publication-title: IEEE Trans. PAMI – volume: 21 start-page: 334 year: 2011 end-page: 371 article-title: Band‐limited localized Parseval frames and Besov spaces on compact homogeneous manifolds publication-title: J. Geom. Anal. – year: 2011 – volume: 2 start-page: 1019 year: 1999 end-page: 1025 article-title: Hierarchical models of object recognition in cortex publication-title: Nat. Neurosci. – volume: 4 start-page: 1201 year: 2005 end-page: 1232 article-title: Multiscale representations for manifold‐valued data publication-title: Multiscale Model. Simul. – volume: 127 start-page: 79 year: 1971 end-page: 183 article-title: Fourier integral operators. I publication-title: Acta Math. – volume: 37 year: 1992 – start-page: 162 year: 2009 end-page: 170 article-title: On invariance in hierarchical models publication-title: Advances in Neural Information Processing Systems 22 – volume: 1 start-page: 1 year: 1985 end-page: 56 article-title: Opérateurs de Calderón‐Zygmund, fonctions para‐accrétives et interpolation. (French) [Calderón‐Zygmund operators, para‐accretive functions and interpolation] publication-title: Rev. Mat. Iberoamericana – year: 2004 – year: 1995 – year: 1970 – year: 1991 – start-page: 1561 year: 2011 end-page: 1566 article-title: Classification with scattering operators publication-title: 2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) – start-page: 253 end-page: 256 – year: 1993 – volume: 37 start-page: 17 year: 2005 end-page: 59 article-title: Local geometry of deformable templates publication-title: SIAM J. Math. Anal. – volume: 34 start-page: 683 year: 1998 end-page: 696 article-title: On contraction analysis for nonlinear systems publication-title: Automatica – year: 2012 – ident: e_1_2_1_5_2 article-title: Invariant scattering convolution networks publication-title: IEEE Trans. PAMI – ident: e_1_2_1_15_2 doi: 10.1111/j.1749-6632.1988.tb32998.x – ident: e_1_2_1_14_2 – volume-title: The geometry of physics: an introduction year: 2004 ident: e_1_2_1_7_2 – ident: e_1_2_1_12_2 doi: 10.1109/ISCAS.2010.5537907 – ident: e_1_2_1_17_2 doi: 10.1137/050622729 – start-page: 162 year: 2009 ident: e_1_2_1_3_2 article-title: On invariance in hierarchical models publication-title: Advances in Neural Information Processing Systems 22 – ident: e_1_2_1_22_2 doi: 10.1137/S0036141002404838 – volume-title: General topology year: 1970 ident: e_1_2_1_23_2 – ident: e_1_2_1_6_2 doi: 10.4171/RMI/17 – ident: e_1_2_1_13_2 doi: 10.1016/S0005-1098(98)00019-3 – ident: e_1_2_1_18_2 doi: 10.1038/14819 – ident: e_1_2_1_16_2 doi: 10.1017/CBO9780511609565 – ident: e_1_2_1_19_2 – ident: e_1_2_1_2_2 – ident: e_1_2_1_4_2 doi: 10.1109/CVPR.2011.5995635 – volume-title: Harmonic analysis: real‐variable methods, orthogonality, and oscillatory integrals year: 1993 ident: e_1_2_1_21_2 – ident: e_1_2_1_10_2 – ident: e_1_2_1_11_2 doi: 10.1007/BF02392052 – ident: e_1_2_1_9_2 doi: 10.1007/s12220-010-9150-3 – ident: e_1_2_1_8_2 doi: 10.1090/cbms/079 – ident: e_1_2_1_20_2 doi: 10.1515/9781400881871 |
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| Snippet | This paper constructs translation‐invariant operators on
$\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$, which are Lipschitz‐continuous... This paper constructs translation‐invariant operators on $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}{\bf L}^2({{{\R}}}^d)$ , which are... This paper constructs translation-invariant operators on L2(Rd), which are Lipschitz-continuous to the action of diffeomorphisms. A scattering propagator is a... |
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| Title | Group Invariant Scattering |
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