Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

The Cannon–Thurston map for punctured-surface groups

Abstract: Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.

Group Equivariant Capsule Networks

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.

The free splitting complex of a free group I: Hyperbolicity

Abstract: Given a free group Fn of finite rank n 2, a free splitting over Fn is a minimal, simplicial action of the group Fn on a simplicial tree T such that the stabilizer of each edge of T is the trivial subgroup of Fn . A free splitting is denoted Fn O T , or just T , when the group and its action are understood. Although the tree T is allowed to have vertices of valence 2, there is a unique natural cell structure on T the vertices of which are the points of valence at least 3. We say that T is a k –edge free splitting if k is the number of natural edge orbits, a number which can take on any value from 1 to 3n 3. The equivalence relation amongst free splittings is conjugacy, where two free splittings of Fn are conjugate if there exists an Fn –equivariant homeomorphism between them. See the beginning of Section 1 for the details of these definitions.

Lorentz Group Equivariant Neural Network for Particle Physics

Abstract: We present a neural network architecture that is fully equivariant with respect to transformations under the Lorentz group, a fundamental symmetry of space and time in physics. The architecture is based on the theory of the finite-dimensional representations of the Lorentz group and the equivariant nonlinearity involves the tensor product. For classification tasks in particle physics, we demonstrate that such an equivariant architecture leads to drastically simpler models that have relatively few learnable parameters and are much more physically interpretable than leading approaches that use CNNs and point cloud approaches. The competitive performance of the network is demonstrated on a public classification dataset [27] for tagging top quark decays given energy-momenta of jet constituents produced in proton-proton collisions.