Spatio-spectral properties of the Wavelet Transform provide a useful theoretical framework to investigate the structure of neural networks. A few researchers (Pati & Krishnaprasad, Zhang & Benveniste) have investigated the connection between neural networks and wavelet transforms. However, a number of issues remain unresolved especially when the connection is considered in the multidimensional case. In our work, we resolve these issues by extensions based on some theorems of Daubechies related to wavelet frames and provide a frame-work to analyze local learning in neural-networks.

We also provide a constructive procedure to build networks based on wavelet theory. Moreover, cognizant of the problems usually encountered in practical implementations of these ideas, we develop a heuristic methodology, inspired by similar work in the area of Radial Basis Function (RBF) networks (Moody & Darken, Platt), to build a network sequentially on-line as well as off-line.

We show some connections of our method to some existing methods such as Projection Pursuit Regression (Friedman), Hyper Basis Functions (Poggio & Girosi) and other methods that have been proposed in the literature on neural- networks as well as statistics. In particular, some classes of wavelets can also be derived from the regularization theoretical framework given by Poggio & Girosi.

Finally, we choose direct nonlinear adaptive control to demonstrate the utility of the network in the context of local learning. Stability analysis is carried out within a standard Lyapunov formulation. Simulation studies show the effectiveness of these methods. We compare and contrast these methods with some recent results obtained by other researchers using Back Propagation (Feed-Forward) Networks, and Gaussian Networks.