相关论文: Fast and accurate normalization of vectors and qua…

This paper presents a pedagogical introduction to the issue of how to implement Lorentz transformations in relativistic visualization. The most efficient approach is to use the even geometric algebra in 3+1 spacetime dimensions, or…

A method is presented for fast diagonalization of a 2x2 or 3x3 real symmetric matrix, that is determination of its eigenvalues and eigenvectors. The Euler angles of the eigenvectors are computed. A small computer algebra program is used to…

Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the…

We construct the quaternion algebra [10] "geometrically" by a three dimensional analogue of the classic two dimensional geometric description of the complex field. The algebraic description of the multiplication operation in three…

We present weight normalization: a reparameterization of the weight vectors in a neural network that decouples the length of those weight vectors from their direction. By reparameterizing the weights in this way we improve the conditioning…

Objects' rigid motions in 3D space are described by rotations and translations of a highly-correlated set of points, each with associated $x,y,z$ coordinates that real-valued networks consider as separate entities, losing information.…

This paper presents an accelerated quadrature scheme for the evaluation of layer potentials in three dimensions. Our scheme combines a generic, high order quadrature method for singular kernels called Quadrature by Expansion (QBX) with a…

This article investigates Kak neural networks, which can be instantaneously trained, for complex and quaternion inputs. The performance of the basic algorithm has been analyzed and shown how it provides a plausible model of human perception…

Quaternion symmetry is ubiquitous in the physical sciences. As such, much work has been afforded over the years to the development of efficient schemes to exploit this symmetry using real and complex linear algebra. Recent years have also…

Dual quaternions have gained significant attention due to their wide applications in areas such as multi-agent formation control, 3D motion modeling, and robotics. A fundamental aspect in dual quaternion research involves the projection…

We study the properties of ultrametric matrices aiming to design methods for fast ultrametric matrix-vector multiplication. We show how to encode such a matrix as a tree structure in quadratic time and demonstrate how to use the resulting…

The paper explains how a unit generalized quaternion is used to represent a rotation of a vector in 3-dimensional space. We review of some algebraic properties of generalized quaternions and operations between them and then show their…

The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's…

We show that the core reasons that complex and hypercomplex valued neural networks offer improvements over their real-valued counterparts is the weight sharing mechanism and treating multidimensional data as a single entity. Their algebra…

Iterative methods with certified convergence for the computation of Gauss--Jacobi quadratures are described. The methods do not require a priori estimations of the nodes to guarantee its fourth-order convergence. They are shown to be…

Dual quaternion algebra and its application to robotics have gained considerable interest in the last two decades. Dual quaternions have great geometric appeal and easily capture physical phenomena inside an algebraic framework that is…

The successful training of deep neural networks requires addressing challenges such as overfitting, numerical instabilities leading to divergence, and increasing variance in the residual stream. A common solution is to apply regularization…

A given six dimensional vector represents a 3D straight line in Plucker coordinates if its coordinates satisfy the Klein quadric constraint. In many problems aiming to find the Plucker coordinates of lines, noise in the data and other…

The normalization of query and key vectors is an essential part of the Transformer architecture. It ensures that learning is stable regardless of the scale of these vectors. Some normalization approaches are available. In this preliminary…

Motivated by a model in quantum computation we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the…