Topic
Equivariant map
About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.
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Proceedings Article•
01 Jun 2018TL;DR: Group equivariant capsule networks as mentioned in this paper use a generic routing by agreement algorithm defined on elements of a group and prove that equivariance of output pose vectors, as well as output activations, hold under certain conditions.
Abstract: We present group equivariant capsule networks, a framework to introduce guaranteed equivariance and invariance properties to the capsule network idea. Our work can be divided into two contributions. First, we present a generic routing by agreement algorithm defined on elements of a group and prove that equivariance of output pose vectors, as well as invariance of output activations, hold under certain conditions. Second, we connect the resulting equivariant capsule networks with work from the field of group convolutional networks. Through this connection, we provide intuitions of how both methods relate and are able to combine the strengths of both approaches in one deep neural network architecture. The resulting framework allows sparse evaluation of the group convolution operator, provides control over specific equivariance and invariance properties, and can use routing by agreement instead of pooling operations. In addition, it is able to provide interpretable and equivariant representation vectors as output capsules, which disentangle evidence of object existence from its pose.
73 citations
TL;DR: In this article, a Lie-theoretic construction of a conjectural mirror family associated to a general flag variety G/P was given, and it recovered the Peterson variety presentation for the T -equivariant quantum cohomology rings q H T ∗ (G / P ) ( q ) with quantum parameters inverted.
73 citations
TL;DR: In this article, it was shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism.
Abstract: It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and fibrations.
73 citations
TL;DR: In this article, a geometric background for representation with the highest weight of the Virasoro algebra is given for representation space consisting of holomorphic sections of an analytic line bundle over the manifold M = Diff+S1/Rot S1 or over its factor manifold M1 = Diff +S 1/PSL (2, ).
73 citations
Posted Content•
TL;DR: Gauge Equivariant Mesh CNNs are proposed which generalize GCNs to apply anisotropic gauge equivariant kernels and introduce a geometric message passing scheme defined by parallel transporting features over mesh edges.
Abstract: A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs). Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole. We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Since the resulting features carry orientation information, we introduce a geometric message passing scheme defined by parallel transporting features over mesh edges. Our experiments validate the significantly improved expressivity of the proposed model over conventional GCNs and other methods.
72 citations