52 research outputs found
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
Inability of spatial transformations of CNN feature maps to support invariant recognition
A large number of deep learning architectures use spatial transformations of
CNN feature maps or filters to better deal with variability in object
appearance caused by natural image transformations. In this paper, we prove
that spatial transformations of CNN feature maps cannot align the feature maps
of a transformed image to match those of its original, for general affine
transformations, unless the extracted features are themselves invariant. Our
proof is based on elementary analysis for both the single- and multi-layer
network case. The results imply that methods based on spatial transformations
of CNN feature maps or filters cannot replace image alignment of the input and
cannot enable invariant recognition for general affine transformations,
specifically not for scaling transformations or shear transformations. For
rotations and reflections, spatially transforming feature maps or filters can
enable invariance but only for networks with learnt or hardcoded rotation- or
reflection-invariant featuresComment: 22 pages, 3 figure
Learning Irreducible Representations of Noncommutative Lie Groups
Recent work has constructed neural networks that are equivariant to
continuous symmetry groups such as 2D and 3D rotations. This is accomplished
using explicit group representations to derive the equivariant kernels and
nonlinearities. We present two contributions motivated by frontier applications
of equivariance beyond rotations and translations. First, we relax the
requirement for explicit Lie group representations, presenting a novel
algorithm that finds irreducible representations of noncommutative Lie groups
given only the structure constants of the associated Lie algebra. Second, we
demonstrate that Lorentz-equivariance is a useful prior for object-tracking
tasks and construct the first object-tracking model equivariant to the
Poincar\'e group.Comment: 15 pages, 5 figure
Quantum finite automata and quiver algebras
National Science FoundationPublished versio
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