A Theory of Neural Computation with Clifford Algebras

The present thesis introduces Clifford Algebra as a framework for neural computation. Neural computation with Clifford algebras is model-based. This principle is established by constructing Clifford algebras from quadratic spaces. Then the subspace grading inherent to any Clifford algebra is introduced. The above features of Clifford algebras are then taken as motivation for introducing the Basic Clifford Neuron (BCN). As a second type of Clifford neuron the Spinor Clifford Neuron is presented. A systematic basis for Clifford neural computation is provided by the important notions of isomorphic Clifford neurons and isomorphic representations. After the neuron level is established, the discussion continues with (Spinor) Clifford Multilayer Perceptrons. First, (Spinor) Clifford Multilayer Perceptrons with real-valued activation functions ((S)CMLPs) are studied. A generic Backpropagation algorithm for CMLPs is derived. Also, universal approximation theorems for (S)CMLPs are presented. Finally, CMLPs with Clifford-valued activation functions are studied

SO(4)SO(4) Landau Models and Matrix Geometry

We develop an in-depth analysis of the $SO(4)$ Landau models on $S^3$ in the $SU(2)$ monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the $SU(2)$ monopole are introduced to provide a concrete coordinate representation of $SO(4)$ operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the $SO(4)$ covariance of the eigenfunctions. With the spin connection of $S^3$, we construct an $SO(4)$ invariant Weyl-Landau operator and analyze its eigenvalue problem with explicit form of the eigenstates. The obtained results include the known formulae of the free Weyl operator eigenstates in the free field limit. Other eigenvalue problems of variant relativistic Landau models, such as massive Dirac-Landau and supersymmetric Landau models, are investigated too. With the developed $SO(4)$ technologies, we derive the three-dimensional matrix geometry in the Landau models. By applying the level projection method to the Landau models, we identify the matrix elements of the $S^3$ coordinates as the fuzzy three-sphere. For the non-relativistic model, it is shown that the fuzzy three-sphere geometry emerges in each of the Landau levels and only in the degenerate lowest energy sub-bands. We also point out that Dirac-Landau operator accommodates two fuzzy three-spheres in each Landau level and the mass term induces interaction between them.Comment: 1+59 pages, 8 figures, 1 table, minor corrections, published versio

Fermions in three-dimensional spinfoam quantum gravity

We study the coupling of massive fermions to the quantum mechanical dynamics of spacetime emerging from the spinfoam approach in three dimensions. We first recall the classical theory before constructing a spinfoam model of quantum gravity coupled to spinors. The technique used is based on a finite expansion in inverse fermion masses leading to the computation of the vacuum to vacuum transition amplitude of the theory. The path integral is derived as a sum over closed fermionic loops wrapping around the spinfoam. The effects of quantum torsion are realised as a modification of the intertwining operators assigned to the edges of the two-complex, in accordance with loop quantum gravity. The creation of non-trivial curvature is modelled by a modification of the pure gravity vertex amplitudes. The appendix contains a review of the geometrical and algebraic structures underlying the classical coupling of fermions to three dimensional gravity.Comment: 40 pages, 3 figures, version accepted for publication in GER

A quaternion deterministic monogenic CNN layer for contrast invariance

Deep learning (DL) is attracting considerable interest as it currently achieves remarkable performance in many branches of science and technology. However, current DL cannot guarantee capabilities of the mammalian visual systems such as lighting changes. This paper proposes a deterministic entry layer capable of classifying images even with low-contrast conditions. We achieve this through an improved version of the quaternion monogenic wavelets. We have simulated the atmospheric degradation of the CIFAR-10 and the Dogs and Cats datasets to generate realistic contrast degradations of the images. The most important result is that the accuracy gained by using our layer is substantially more robust to illumination changes than nets without such a layer.The authors would like to thank to CONACYT and Barcelona supercomputing Center. Sebastián Salazar-Colores (CVU 477758) would like to thank CONACYT (Consejo Nacional de Ciencia y Tecnología) for the financial support of his PhD studies under Scholarship 285651. Ulises Moya and Ulises Cortés are member of the Sistema Nacional de Investigadores CONACyT.Peer ReviewedPostprint (author's final draft

51 Applications of Geometric Algebra in Robot Vision

Abstract In this tutorial paper we will report on our experience in the use of geometric algebra (GA) in robot vision. The results could be reached in a long term research programme on modelling the perception-action cycle within geometric algebra. We will pick up three important applications from image processing, pattern recognition and computer vision. By presenting the problems and their solutions from an engineering point of view, the intention is to stimulate other applications of $\mathrm{G}\mathrm{A}$

Maximum principle for the regularized Schrödinger operator

In this paper we present analogues of the maximum principle and of some parabolic inequalities for the regularized time-dependent Schrödinger operator on open manifolds using Günter derivatives. Moreover, we study the uniqueness of bounded solutions for the regularized Schrödinger-Günter problem and obtain the corresponding fundamental solution. Furthermore, we present a regularized Schrödinger kernel and prove some convergence results. Finally, we present an explicit construction for the fundamental solution to the Schrödinger-Günter problem on a class of conformally flat cylinders and tori

Struktureller Bias in neuronalen Netzen mittels Clifford-Algebren

Im Rahmen dieser Arbeit wird ein generisches Approximierungsmodell aufgestellt, das unter anderem klassische neuronale Architekturen umfaßt. Die allgemeine Rolle von a priori Wissen bei der Modellierung wird untersucht. Speziell werden Clifford-Algebren bei dem Entwurf von neuronalen Netzen als Träger struktureller Information eingesetzt. Diese Wahl wird durch die Eigenschaft von Clifford-Algebren motiviert, geometrische Entitäten sowie deren Transformationen auf eine effiziente Art darstellen bzw. berechnen zu können. Neue neuronale Architekturen, die im Vergleich zu klassischen Ansätzen höhere Effizienz aufweisen, werden entwickelt und zur Lösung von verschiedenen Aufgaben in Bildverarbeitung, Robotik und Neuroinformatik allgemein eingesetzt

On symplectic 4-manifolds and contact 5-manifolds

In dieser Arbeit werden einige Aussagen über symplektische Strukturen auf 4-dimensionalen Mannigfaltigkeiten und Kontaktstrukturen auf 5-dimensionalen Mannigfaltigkeiten bewiesen. Wir untersuchen zunächst den Zusammenhang zwischen dem symplektischen und dem holomorphen Minimalitätsbegriff für Kählerflächen. Außerdem beweisen wir ein Resultat über die Irreduzibilität minimaler, einfach-zusammenhängender symplektischer 4- Mannigfaltigkeiten unter zusammenhängender Summe und eine Aussage über die konformen Systolen symplektischer 4-Mannigfaltigkeiten. Als nächstes betrachten wir die Konstruktion von differenzierbaren 4-dimensionalen Mannigfaltigkeiten durch die verallgemeinerte Fasersumme. Für den Fall, dass die Summation entlang eingebetteter Flächen mit trivialem Normalenbündel erfolgt, werden die ganzzahligen Homologiegruppen und im symplektischen Fall auch die kanonische Klasse der Fasersumme berechnet. Wir betrachten verschiedene Anwendungen, insbesondere hinsichtlich der Geographie einfach-zusammenhängender symplektischer 4-Mannigfaltigkeiten, deren kanonische Klasse durch eine vorgegebene natürliche Zahl teilbar ist. Wir zeigen auch, dass man mit geeigneten verzweigten Überlagerungen von komplexen Flächen vom allgemeinen Typ einfach-zusammenhängende algebraische Flächen konstruieren kann, deren kanonische Klasse eine vorgegebene Teilbarkeit besitzt. Im zweiten Teil der Arbeit betrachten wir die Boothby-Wang Konstruktion von Kontaktstrukturen auf Kreisbündeln über symplektischen Mannigfaltigkeiten. Zusammen mit den Resultaten über Geographie aus dem ersten Teil der Arbeit zeigen wir, dass es auf bestimmten einfach-zusammenhängenden 5-Mannigfaltigkeiten Kontaktstrukturen gibt, die nicht äquivalent sind, aber die in derselben (nicht-trivialen) Homotopieklasse von Fast-Kontaktstrukturen liegen