Elementary derivative tasks and neural net multiscale analysis of tasks

Formal neurons implementing wavelets have been shown to build nets that are able to approximate any multidimensional task. In this paper, we use a finite number of formal neurons implementing elementary tasks such as "sombrero" responses or even simpler "window" responses, with adjustable widths. We show this to provide a reasonably efficient, practical and robust, multifrequency analysis of tasks. The translation degree of freedom of wavelets is shown to be unnecessary. A training algorithm, optimizing the output task with respect to the widths of the responses, reveals two distinct training modes. The first mode keeps the formal neurons distinct. The other mode induces some of the formal neurons to become identical, with output weights of equal strengths but opposite signs. Hence this latter mode promotes tasks that are derivatives of the elementary tasks with respect to the width parameter. Such results, obtained from optimizations with respect to a width parameter, can be generalized for any other parameters of the elementary tasks.