1,450 research outputs found
Differential Geometry of Group Lattices
In a series of publications we developed "differential geometry" on discrete
sets based on concepts of noncommutative geometry. In particular, it turned out
that first order differential calculi (over the algebra of functions) on a
discrete set are in bijective correspondence with digraph structures where the
vertices are given by the elements of the set. A particular class of digraphs
are Cayley graphs, also known as group lattices. They are determined by a
discrete group G and a finite subset S. There is a distinguished subclass of
"bicovariant" Cayley graphs with the property that ad(S)S is contained in S.
We explore the properties of differential calculi which arise from Cayley
graphs via the above correspondence. The first order calculi extend to higher
orders and then allow to introduce further differential geometric structures.
Furthermore, we explore the properties of "discrete" vector fields which
describe deterministic flows on group lattices. A Lie derivative with respect
to a discrete vector field and an inner product with forms is defined. The
Lie-Cartan identity then holds on all forms for a certain subclass of discrete
vector fields.
We develop elements of gauge theory and construct an analogue of the lattice
gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear
connections are considered and a simple geometric interpretation of the torsion
is established.
By taking a quotient with respect to some subgroup of the discrete group,
generalized differential calculi associated with so-called Schreier diagrams
are obtained.Comment: 51 pages, 11 figure
CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters
The rise of graph-structured data such as social networks, regulatory
networks, citation graphs, and functional brain networks, in combination with
resounding success of deep learning in various applications, has brought the
interest in generalizing deep learning models to non-Euclidean domains. In this
paper, we introduce a new spectral domain convolutional architecture for deep
learning on graphs. The core ingredient of our model is a new class of
parametric rational complex functions (Cayley polynomials) allowing to
efficiently compute spectral filters on graphs that specialize on frequency
bands of interest. Our model generates rich spectral filters that are localized
in space, scales linearly with the size of the input data for
sparsely-connected graphs, and can handle different constructions of Laplacian
operators. Extensive experimental results show the superior performance of our
approach, in comparison to other spectral domain convolutional architectures,
on spectral image classification, community detection, vertex classification
and matrix completion tasks
Differential Calculi on Quantum Spaces determined by Automorphisms
If the bimodule of 1-forms of a differential calculus over an associative
algebra is the direct sum of 1-dimensional bimodules, a relation with
automorphisms of the algebra shows up. This happens for some familiar quantum
space calculi.Comment: 7 pages, Proceedings of XIIIth International Colloquium Integrable
Systems and Quantum Group
Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups
In the present paper, we develop geometric analytic techniques on Cayley
graphs of finitely generated abelian groups to study the polynomial growth
harmonic functions. We develop a geometric analytic proof of the classical
Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic
functions on lattices \mathds{Z}^n that does not use a representation formula
for harmonic functions. We also calculate the precise dimension of the space of
polynomial growth harmonic functions on finitely generated abelian groups.
While the Cayley graph not only depends on the abelian group, but also on the
choice of a generating set, we find that this dimension depends only on the
group itself.Comment: 15 pages, to appear in Ann. Global Anal. Geo
Dirac operators and spectral triples for some fractal sets built on curves
We construct spectral triples and, in particular, Dirac operators, for the
algebra of continuous functions on certain compact metric spaces. The triples
are countable sums of triples where each summand is based on a curve in the
space. Several fractals, like a finitely summable infinite tree and the
Sierpinski gasket, fit naturally within our framework. In these cases, we show
that our spectral triples do describe the geodesic distance and the Minkowski
dimension as well as, more generally, the complex fractal dimensions of the
space. Furthermore, in the case of the Sierpinski gasket, the associated
Dixmier-type trace coincides with the normalized Hausdorff measure of dimension
.Comment: 48 pages, 4 figures. Elementary proofs omitted. To appear in Adv.
Mat
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