Matrix Formulas for Semilinear Backpropagation
Résumé
The present article is a direct continuation of, and should be read concurrently with, reference [1]. In that paper formulas were obtained for the gradient of the quadratic error of a general differentiable multilayer neural network defined over Hilbert spaces. The formulas belong to the class of backpropagation procedures that express the gradient in terms of the output errors and the transpose partial derivatives of the layers. Here we first develop a mathematical framework for semilinear perceptron neural networks using the natural correspondence between the transpose partial derivatives and the transpose partial Jacobians of the layers, such Jacobians being expressed in terms of the perceptron data as equations (11) and (14). The final gradient formulas for perceptron networks are presented as equations (16) and (17). Section §14 contains figures with captions to complement the discussion.
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