Group Invariant Scattering

This paper constructs translation-invariant operators on 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 L-2(R-d) and on L-2(SO(d)) defines a translation- and rotation-invariant scattering on L-2(R-d). (c) 2012 Wiley Periodicals, Inc.