On the optimal weight of a perceptron with Gaussian data and arbitrary nonlinearity
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In this correspondence we investigate the solution to the following problem: Find the optimal weighted sum of given signals when the optimality criteria is the expected value of a function of this sum and a given "training" signal. The optimality criteria can be a nonlinear function from a very large family of possible functions. A number of interesting cases fall under this general framework, such as a single layer perceptron with any of the commonly used nonlinearities, the LMS, the LMF or higher moments, or the various sign algorithms. Assuming the signals to be jointly Gaussian we show that the optimal solution, when it exits, is always collinear with the well-known Wiener solution, and only its scaling factor depends on the particular functions chosen. We also present necessary constructive conditions for the existance of the optimal solution.
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