Geometric preprocessing, geometric feedforward neural networks and Clifford support vector machines for visual learning

  • Authors:

  • Eduardo Bayro-Corrochano;Refugio Vallejo;Nancy Arana-Daniel

  • Affiliations:

  • Computer Science Department, CINVESTAV Centro de Investigación y de Estudios Avanzados, GEOVIS Laboratory Apartado Postal 31-438, Plaza la Luna, Guadalajara, Jal. 44550, Mexico;Escuela de Economía, Universidad de Guanajuato, Apartado Postal 533, 36000-Guanajuato, Gto. Mexico;Computer Science Department, CINVESTAV Centro de Investigación y de Estudios Avanzados, GEOVIS Laboratory Apartado Postal 31-438, Plaza la Luna, Guadalajara, Jal. 44550, Mexico

  • Venue:
  • Neurocomputing
  • Year:
  • 2005

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Abstract

This paper shows the design and use of feed-forward neural networks and the support vector machines (SVM) in the coordinate-free mathematical system of the Clifford geometric algebra. We compare the McCulloch-Pitts neuron and the geometric neuron. An interesting case of the geometric neuron is the conformal neuron which can be used for RBF networks and SVM. The paper presents the generalization of the real- and complex-valued multilayer perceptron (MLP) to the Clifford valued multilayer perceptron. The paper studies also the multivector support vector machines (MSVM) which are SVMs for processing multivectors. For that we design kernels involving Clifford products. The resultant kernel resembles a sort of polynomial kernel using a multivector representation. In the context of SVMs an important contribution of the paper is the generalization of the real- and complex-valued SVM classifiers over the hyper-complex numbers. This Clifford valued SVM accepts multiple multivector inputs and it is a multi-class classifier. For the preprocessing the authors introduce a promising geometric method utilizing Clifford moments. This method is applied together with geometric MLPs for tasks of 2D pattern recognition. The experimental part shows applications of SVM using the conformal neuron and Clifford kernels. We include challenging applications of the Clifford SVM classifier for nonlinear separable problems. The authors believe that the use of the MLPs and SVMs in the geometric algebra framework expands their sphere of applicability for multivector learning in graded spaces.