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A General Theory of Equivariant CNNs on Homogeneous Spaces | | Taco Cohen
; Mario Geiger
; Maurice Weiler
; | | Date: |
5 Nov 2018 | | Abstract: | Group equivariant convolutional neural networks (G-CNNs) have recently
emerged as a very effective model class for learning from signals in the
context of known symmetries. A wide variety of equivariant layers has been
proposed for signals on 2D and 3D Euclidean space, graphs, and the sphere, and
it has become difficult to see how all of these methods are related, and how
they may be generalized.
In this paper, we present a fairly general theory of equivariant
convolutional networks. Convolutional feature spaces are described as fields
over a homogeneous base space, such as the plane $mathbb{R}^2$, sphere $S^2$
or a graph $mathcal{G}$. The theory enables a systematic classification of all
existing G-CNNs in terms of their group of symmetry, base space, and field type
(e.g. scalar, vector, or tensor field, etc.).
In addition to this classification, we use Mackey theory to show that
convolutions with equivariant kernels are the most general class of equivariant
maps between such fields, thus establishing G-CNNs as a universal class of
equivariant networks. The theory also explains how the space of equivariant
kernels can be parameterized for learning, thereby simplifying the development
of G-CNNs for new spaces and symmetries. Finally, the theory introduces a rich
geometric semantics to learned feature spaces, thus improving interpretability
of deep networks, and establishing a connection to central ideas in mathematics
and physics. | | Source: | arXiv, 1811.2017 | | Services: | Forum | Review | PDF | Favorites | |
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