Equivariant map

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

On Learning Sets of Symmetric Elements

Abstract: Learning from unordered sets is a fundamental learning setup, recently attracting increasing attention. Research in this area has focused on the case where elements of the set are represented by feature vectors, and far less emphasis has been given to the common case where set elements themselves adhere to their own symmetries. That case is relevant to numerous applications, from deblurring image bursts to multi-view 3D shape recognition and reconstruction. In this paper, we present a principled approach to learning sets of general symmetric elements. We first characterize the space of linear layers that are equivariant both to element reordering and to the inherent symmetries of elements, like translation in the case of images. We further show that networks that are composed of these layers, called Deep Sets for Symmetric Elements (DSS) layers, are universal approximators of both invariant and equivariant functions, and that these networks are strictly more expressive than Siamese networks. DSS layers are also straightforward to implement. Finally, we show that they improve over existing set-learning architectures in a series of experiments with images, graphs, and point-clouds.

On equivariant vector bundles on an almost homogeneous variety

Abstract: Let k be an algebraically closed field of arbitrary characteristic. Let T be an n-dimensional algebraic torus, i.e. T = Gm × · · · × Gm n-times), where Gm = Spec (k[t, t-1]) is the multiplicative group.

Equivariant Weierstrass Preparation and Values of L-functions at Negative Integers

Abstract: We apply an equivariant version of the p-adic Weierstrass Preparation Theorem in the context of possible non-commutative gen- eralizations of the power series of Deligne and Ribet. We then con- sider CM abelian extensions of totally real fields and by combining our earlier considerations with the known validity of the Main Con- jecture of Iwasawa theory we prove, modulo the conjectural vanishing of certain µ-invariants, a (corrected version of a) conjecture of Snaith and the 'rank zero component' of Kato's Generalized Iwasawa Main Conjecture for Tate motives of strictly positive weight. We next use the validity of this case of Kato's conjecture to prove a conjecture of Chinburg, Kolster, Pappas and Snaith and also to compute ex- plicitly the Fitting ideals of certain naturalcohomology groups in terms of the values of Dirichlet L-functions at negative integers. This computation improves upon results of Cornacchia and Ostvaer, of Kurihara and of Snaith, and, modulo the validity of a certain aspect of the Quillen-Lichtenbaum Conjecture, also verifies a finer and more general version of a well known conjecture of Coates and Sinnott.

The Dirac operator on compact quantum groups

Abstract: For the q-deformation G_q, 0