Group Invariant Scattering

• Published in: Communications on pure and applied mathematics Vol. 65; no. 10; pp. 1331 - 1398
• Main Author: Mallat, Stéphane
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.10.2012; John Wiley and Sons, Limited
• Subjects: Expected values; Lie groups; Mathematical functions; Scattering; Wavelet transforms
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.21413

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.