Kon 9720145 The investigator studies neural network architectures and applications of wavelet techniques to investigate neural networks' complexity. Wavelets have been established as a rich and useful family of expansion functions. The investigator studies further the recovery of functions (in particular representations of visual images) from their wavelet transforms. Issues of stability and complexity, which have not up to now been addressed in his proof of the Marr conjecture and related analysis of the Mallat conjecture, are studied. An important current question regards the complexity of such networks (i.e., their essential size) for the completion of desired tasks. The investigator studies two types, so-called functional and logical networks. Functional networks have received a good deal of attention, and a coherent theory has established that they are essentially orthogonal (or more general) expansion engines. The homology between the structure of networks and expansion tasks has allowed establishing the connection of wavelet convergence results with network complexity issues. The investigator examines the class of so-called logical networks as a needed completion of available network architectures for the execution of intelligent tasks. In addition he works to show that wavelet-based neural networks achieve lower bounds on complexities of neural nets for given tasks. These results move toward a general complexity theory for neural nets on the order of current computational complexity theory for serial and parallel computer architectures. Such a complexity theory is expected to be a hybrid of current discrete and continuous computational complexity theories. The global purpose of this project is a mathematical study of neural network architectures that implement some of the theoretical complexity results that the investigator obtains. Neural networks as models of parallel distributed computing are currently the leading architectures holdi ng a promise of artificially emulating intelligent systems, as has been indicated in many of their current applications (including mortgage decisions, commercial stock market analysis applications, chemical and thermal homeostasis control systems, satellite image analysis, etc.). A major unanswered question in the development of such systems is the fundamental issue of how large a network needs to be in order to perform specific intelligent functions. One type of task that current so-called "functional" neural architectures have difficulty in dealing with is artificial visual recognition and related tasks involved in the general area of robotics. This difficulty seems to be an inherent part of the functional neural architectures under current study, and the investigator develops architectures involving so-called "logical" components, which act essentially as algorithmic engines. In particular such network architectures are necessary for artificial vision tasks, and prototypes of such tasks are simulated computationally with the aid of graduate students working on the project. Wavelets are currently considered to be one of the most useful tools for representing the types of input-output functions implemented in neural networks. A more general question regarding the complexity and size of neural networks accomplishing real-world tasks is addressed through application of wavelet techniques to network construction. In particular, functional neural networks may achieve their optimal performance using wavelets as activation functions. There is a larger question here regarding whether wavelet techniques are the best possible for the implementation of functional neural network architectures, which is a conjecture the investigator has made and investigates. The computational aspects of the project are aided by associated groups at Howard University and Bryn Mawr College, the Howard group involving a number of graduate students.