Related papers: Geometric operations implemented by conformal geom…
The concept of viewing graph solvability has gained significant interest in the context of structure-from-motion. A viewing graph is a mathematical structure where nodes are associated to cameras and edges represent the epipolar geometry…
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…
Euclidean deep learning is often inadequate for addressing real-world signals where the representation space is irregular and curved with complex topologies. Interpreting the geometric properties of such feature spaces has become paramount…
Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is…
Image-based 3D object modeling refers to the process of converting raw optical images to 3D digital representations of the objects. Very often, such models are desired to be dimensionally true, semantically labeled with photorealistic…
$Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose…
In this paper geometry is studied with a novel approach. Every geometrical object is defined as a symbol which satisfies some properties. These symbols are then coded into a class of numbers which are named here as many dots numbers (MDN).…
Any procedure applied to data, and any quantity derived from data, is required to respect the nature and symmetries of the data. This axiom applies to refinement procedures and multiresolution transforms as well as to more basic operations…
Evaluating the quality of learned representations without relying on a downstream task remains one of the challenges in representation learning. In this work, we present Geometric Component Analysis (GeomCA) algorithm that evaluates…
In a recent paper, a new conformally flat metric was introduced, describing an expanding scalar field in a spherically symmetric geometry. The spacetime can be interpreted as a Schwarzschild-like model with an apparent horizon surrounding…
Solving geometric tasks involving point clouds by using machine learning is a challenging problem. Standard feed-forward neural networks combine linear or, if the bias parameter is included, affine layers and activation functions. Their…
Why and how that deep learning works well on different tasks remains a mystery from a theoretical perspective. In this paper we draw a geometric picture of the deep learning system by finding its analogies with two existing geometric…
Mathematical objects are generally abstract and not very approachable. Illustrations and interactive visualizations help both students and professionals to comprehend mathematical material and to work with it. This approach lends itself…
Geometric algebra is a powerful framework that unifies mathematics and physics. Since its revival in the middle of the 1960s by David Hestenes, it attracts great attention and has been exploited in many fields such as physics, computer…
Human cognition spans perception, memory, intuitive judgment, deliberative reasoning, action selection, and social inference, yet these capacities are often explained through distinct computational theories. Here we present a unified…
Space-Time in general relativity is a dynamical entity because it is subject to the Einstein field equations. The space-time metric provides different geometrical structures: conformal, volume, projective and linear connection. A deep…
We introduce a new formalism and a number of new results in the context of geometric computational vision. The classical scope of the research in geometric computer vision is essentially limited to static configurations of points and lines…
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 quantities we are interested in predicting are geometric tensors; we refer to this class of problems as geometric prediction. Attempts to perform geometric prediction in real-world scenarios have been limited to approximating them…
This paper explores the application of geometric algebra to Galilean spacetime and its physical implications. We introduce the Galilean Spacetime Algebra (GSTA), a five-dimensional conformal geometric algebra (CGA) generated by a specific…