相关论文: Geometric Algebras
In this expository paper we want to give a brief introduction, with few key references for further reading, to the inner functioning of the new and successfull algorithms of Deep Learning and Geometric Deep Learning with a focus on Graph…
The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are…
In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating…
Studies on time and memory costs of products in geometric algebra have been limited to cases where multivectors with multiple grades have only non-zero elements. This allows to design efficient algorithms for a generic purpose; however, it…
In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to…
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…
A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it it shown how first and second order efficient estimators can be constructed,…
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
In this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the…
Many statistical models are algebraic in that they are defined by polynomial constraints or by parameterizations that are polynomial or rational maps. This opens the door for tools from computational algebraic geometry. These tools can be…
Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if…
In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.
Designing software systems for Geometric Computing applications can be a challenging task. Software engineers typically use software abstractions to hide and manage the high complexity of such systems. Without the presence of a unifying…
Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…
We take a categorical approach to describe ternary derivations and ternary automorphisms of triangular algebras. New classes of automorphisms and derivations of triangular algebras are also introduced and studied.
We introduce the notion of rank of multivector in Clifford geometric algebras of arbitrary dimension without using the corresponding matrix representations and using only geometric algebra operations. We use the concepts of characteristic…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
This paper first gives a brief overview over some interesting descriptions of conic sections, showing formulations in the three geometric algebras of Euclidean spaces, projective spaces, and the conformal model of Euclidean space. Second…