G. Ferrari-Trecate and G. De Nicolao. Regularization networks for inverse problems: A state-space approach. Automatica , 39:669--676, 2003.
Linear inverse problems with discrete data are equivalent to the estimationof the continuous-time input of a linear dynamical system from samples of its output.The solution obtained by means of regularization theory has the structure of a neural networksimilar to classical RBF networks.However, the basisfunctions depend in a nontrivial way on the specific linear operator to be inverted and theadopted regularization strategy. By resorting to the Bayesian interpretation ofregularization, we show that such networks can be implemented rigorously and efficientlywhenever the linear operator admits a state-space representation. An analytic expression isprovided for the basis functions as well as for the entries of the matrix of the linearsystem used to compute the weights. Moreover, the weights can be computed in $O(N)$operations by a suitable algorithm based on Kalman filtering. The results are illustratedthrough a deconvolution problem where the spontaneous secretory rate of Luteinizing Hormone(LH) of the hypophisis is reconstructed from measurements of plasma LH concentrations.