ON THE OPTIMAL WEIGHT VECTOR OF A PERCEPTRON WITH GAUSSIAN DATA AND ARBITRARY NONLINEARITY

In this correspondence we investigate the solution to the following pr oblem: Find the optimal weighted sum of given signals when the optimal ity criteria is the expected value of a function of this sum and a giv en ''training'' signal. The optimality criteria can be a nonlinear fun ction from a very large family of possible functions. A number of inte resting cases fall under this general framework, such as a single laye r perceptron with any of the commonly used nonlinearities, the LMS, th e LMF or higher moments, or the various sign algorithms.Assuming the s ignals 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. W e also present necessary constructive conditions for the existance of the optimal solution.