  Wavelet scattering networks, which are convolutional neural networks (CNNs)
with fixed filters and weights, are promising tools for image analysis.
Imposing symmetry on image statistics can improve human interpretability, aid
in generalization, and provide dimension reduction. In this work, we introduce
a fast-to-compute, translationally invariant and rotationally equivariant
wavelet scattering network (EqWS) and filter bank of wavelets (triglets). We
demonstrate the interpretability and quantify the invariance/equivariance of
the coefficients, briefly commenting on difficulties with implementing scale
equivariance. On MNIST, we show that training on a rotationally invariant
reduction of the coefficients maintains rotational invariance when generalized
to test data and visualize residual symmetry breaking terms. Rotation
equivariance is leveraged to estimate the rotation angle of digits and
reconstruct the full rotation dependence of each coefficient from a single
angle. We benchmark EqWS with linear classifiers on EMNIST and CIFAR-10/100,
introducing a new second-order, cross-color channel coupling for the color
images. We conclude by comparing the performance of an isotropic reduction of
the scattering coefficients and RWST, a previous coefficient reduction, on an
isotropic classification of magnetohydrodynamic simulations with astrophysical
relevance.
