604 research outputs found
Learning SO(3) Equivariant Representations with Spherical CNNs
We address the problem of 3D rotation equivariance in convolutional neural
networks. 3D rotations have been a challenging nuisance in 3D classification
tasks requiring higher capacity and extended data augmentation in order to
tackle it. We model 3D data with multi-valued spherical functions and we
propose a novel spherical convolutional network that implements exact
convolutions on the sphere by realizing them in the spherical harmonic domain.
Resulting filters have local symmetry and are localized by enforcing smooth
spectra. We apply a novel pooling on the spectral domain and our operations are
independent of the underlying spherical resolution throughout the network. We
show that networks with much lower capacity and without requiring data
augmentation can exhibit performance comparable to the state of the art in
standard retrieval and classification benchmarks.Comment: Camera-ready. Accepted to ECCV'18 as oral presentatio
A General Theory of Equivariant CNNs on Homogeneous Spaces
We present a general theory of Group equivariant Convolutional Neural
Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere.
Feature maps in these networks represent fields on a homogeneous base space,
and layers are equivariant maps between spaces of fields. The theory enables a
systematic classification of all existing G-CNNs in terms of their symmetry
group, base space, and field type. We also consider a fundamental question:
what is the most general kind of equivariant linear map between feature spaces
(fields) of given types? Following Mackey, we show that such maps correspond
one-to-one with convolutions using equivariant kernels, and characterize the
space of such kernels
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