Group Invariant Scattering

TLDR

A scattering propagator is a path‑ordered product of nonlinear, noncommuting operators that compute the modulus of a wavelet transform. The paper constructs translation‑invariant operators on L²(ℝᵈ) that are Lipschitz‑continuous with respect to diffeomorphisms. These operators are built via a windowed scattering transform that is Lipschitz‑continuous to C² diffeomorphisms, extended to L²(G) for compact Lie groups, and combined with a scattering on L²(SO(d)) to achieve translation and rotation invariance. As the window size grows, the transform converges to a translation‑invariant wavelet scattering, whose coefficients represent stationary processes and discriminate processes with identical power spectra through high‑order moments. © 2012 Wiley Periodicals, Inc.

Abstract

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 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.