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

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

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