Representation of Functional Data in Neural Networks

Functional Data Analysis (FDA) is an extension of traditional data analysis to functional data, for example spectra, temporal series, spatio-temporal images, gesture recognition data, etc. Functional data are rarely known in practice; usually a regular or irregular sampling is known. For this reason, some processing is needed in order to benefit from the smooth character of functional data in the analysis methods. This paper shows how to extend the Radial-Basis Function Networks (RBFN) and Multi-Layer Perceptron (MLP) models to functional data inputs, in particular when the latter are known through lists of input-output pairs. Various possibilities for functional processing are discussed, including the projection on smooth bases, Functional Principal Component Analysis, functional centering and reduction, and the use of differential operators. It is shown how to incorporate these functional processing into the RBFN and MLP models. The functional approach is illustrated on a benchmark of spectrometric data analysis.Comment: Also available online from: http://www.sciencedirect.com/science/journal/0925231

Free Loop Spaces and Cyclohedra

In this note we introduce an action of cyclohedra on the free loop space. We will then discuss how this action can be used for an appropriate recognition principle in the same manner as the action of Stasheff's associahedra can be used to recognize based loop spaces. We will also interpret one result of R.L. Cohen as an approximation theorem, in the spirit of Beck and May, for free loop spaces.Comment: 7 pages, LaTeX 2.0