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Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of…
The purpose of this book is to give an exposition of geometry, from a point of view which complements Klein's Erlangen program. The emphasis is on extending the classical Euclidean geometry to the finite case, but it goes beyond that. After…
The ever-increasing demand for 3D modeling in the emerging immersive applications has made point clouds an essential class of data for 3D image and video processing. Tree based structures are commonly used for representing point clouds…
Accurate and efficient physical simulations are essential in science and engineering, yet traditional numerical solvers face significant challenges in computational cost when handling simulations across dynamic scenarios involving complex…
GeoGebra is an open source mathematics education software tool being used in thousands of schools worldwide. Since version 4.2 (December 2012) it supports symbolic computation of locus equations as a result of joint effort of mathematicians…
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…
We attach the degenerate signature (n,0,1) to the projectivized dual Grassmann algebra over R(n+1). We explore the use of the resulting Clifford algebra as a model for euclidean geometry. We avoid problems with the degenerate metric by…
The remarkable growth and significant success of machine learning have expanded its applications into programming languages and program analysis. However, a key challenge in adopting the latest machine learning methods is the representation…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
This text is a support for different courses of the master of Mechanics of the University Paris-Saclay. The content of this text is an introduction, for graduate students, to tensor algebra and analysis. Far from being exhaustive, the text…
This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…
With the growth of machine learning algorithms with geometry primitives, a high-efficiency library with differentiable geometric operators are desired. We present an optimized Differentiable Geometry Algorithm Library (DGAL) loaded with…
The line geometric model of 3-D projective geometry has the nice property that the Lie algebra sl(4) of 3-D projective transformations is isomorphic to the bivector algebra of CL(3,3), and line geometry is closely related to the classical…
A hard hadron-collider event is treated here as a single geometric object - the kinematics and the discrete object-type labels of all reconstructed final-state particles encoded in one multivector $\evMV\in\Cl(1,3)\otimes\Vflav$ - rather…
The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…
GeoGebra is an interactive geometry, algebra, statistics, and calculus application designed for teaching and learn-ing math, science, and engineering. Its dynamic interface allows its users to accurately and interactively visualize their…
Computational Intelligence (CI) techniques have shown great potential as a surrogate model of expensive physics simulation, with demonstrated ability to make fast predictions, albeit at the expense of accuracy in some cases. For many…
Point cloud registration aims to provide estimated transformations to align point clouds, which plays a crucial role in pose estimation of various navigation systems, such as surgical guidance systems and autonomous vehicles. Despite the…
This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric…
Constructive-deductive method for plane Euclidean geometry is proposed and formalized within Coq Proof Assistant. This method includes both postulates that describe elementary constructions by idealized geometric tools (pencil, straightedge…