Parameter Estimation of Sigmoid Superpositions: Dynamical System Approach

Superposition of sigmoid function over a finite time interval is shown to be equivalent to the linear combination of the solutions of a linearly parameterized system of logistic differential equations. Due to the linearity with respect to the parameters of the system, it is possible to design an effective procedure for parameter adjustment. Stability properties of this procedure are analyzed. Strategies shown in earlier studies to facilitate learning such as randomization of a learning sequence and adding specially designed disturbances during the learning phase are requirements for guaranteeing convergence in the learning scheme proposed.Comment: 30 pages, 7 figure

Approximation with Random Bases: Pro et Contra

In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both randomized and deterministic, for selecting elements from these families that have been shown to ensure the rate of convergence in $L_2$ norm of order $O(1/N)$, where $N$ is the number of elements. We show that both randomized and deterministic procedures are successful if additional information about the families of functions to be approximated is provided. In the absence of such additional information one may observe exponential growth of the number of terms needed to approximate the function and/or extreme sensitivity of the outcome of the approximation to parameters. Implications of our analysis for applications of neural networks in modeling and control are illustrated with examples.Comment: arXiv admin note: text overlap with arXiv:0905.067