A Theory of Networks for Appxoimation and Learning

Learning an input-output mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multi-dimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nolinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. We develop a theoretical framework for approximation based on regularization techniques that leads to a class of three-layer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the well-known Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods such as Parzen windows and potential functions and to several neural network algorithms, such as Kanerva's associative memory, backpropagation and Kohonen's topology preserving map. They also have an interesting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.

A Low Power Wide Dynamic Range Envelope Detector

We report a 75dB, 2.8mW, 100Hz-10kHz envelope detector in a 1.5mm 2.8V CMOS technology. The envelope detector performs input-dc-insensitive voltage-to-currentconverting rectification followed by novel nanopower current-mode peak detection. The use of a subthreshold wide- linear-range transconductor (WLR OTA) allows greater than 1.7Vpp input voltage swings. We show theoretically that this optimal performance is technology-independent for the given topology and may be improved only by spending more power. A novel circuit topology is used to perform 140nW peak detection with controllable attack and release time constants. The lower limits of envelope detection are determined by the more dominant of two effects: The first effect is caused by the inability of amplified high-frequency signals to exceed the deadzone created by exponential nonlinearities in the rectifier. The second effect is due to an output current caused by thermal noise rectification. We demonstrate good agreement of experimentally measured results with theory. The envelope detector is useful in low power bionic implants for the deaf, hearing aids, and speech-recognition front ends. Extension of the envelope detector to higher- frequency applications is straightforward if power consumption is inc

Causal and Teleological Reasoning in Circuit Recognition

This thesis presents a theory of human-like reasoning in the general domain of designed physical systems, and in particular, electronic circuits. One aspect of the theory, causal analysis, describes how the behavior of individual components can be combined to explain the behavior of composite systems. Another aspect of the theory, teleological analysis, describes how the notion that the system has a purpose can be used to aid this causal analysis. The theory is implemented as a computer program, which, given a circuit topology, can construct by qualitative causal analysis a mechanism graph describing the functional topology of the system. This functional topology is then parsed by a grammar for common circuit functions. Ambiguities are introduced into the analysis by the approximate qualitative nature of the analysis. For example, there are often several possible mechanisms which might describe the circuit's function. These are disambiguated by teleological analysis. The requirement that each component be assigned an appropriate purpose in the functional topology imposes a severe constraint which eliminates all the ambiguities. Since both analyses are based on heuristics, the chosen mechanism is a rationalization of how the circuit functions, and does not guarantee that the circuit actually does function. This type of coarse understanding of circuits is useful for analysis, design and troubleshooting.