Vector space methods for sensor fusion problems

In a multiple sensor system, the sensor S_j, j=1,2,... ,N, outputs Y^(j) ∈ ℜ in response to input X ∈ [0,1], according to an unknown probability distribution P_Y(j)\X. The problem is to estimate a fusion function f:ℜ^N→[0,1], based on a training sample, such that the expected square error is minimized over a family of functions .ℱ that constitutes a finite-dimensional vector space. The function f* that exactly minimizes the expected error cannot be computed since the underlying distributions are unknown, and only an approximation f̂ to f* is feasible. We estimate the sample size sufficiently to ensure that an estimator f̂ that minimizes the empirical square error provides a close approximation to f* with a high probability. The advantages of vector space methods are twofold: (1) the sample size estimate is a simple function of the dimensionality of ℱ and (2) the estimate f̂ can be easily computed by the well-known least square methods in polynomial time. The results are applicable to the classical potential function method as well as to a recently proposed class of sigmoidal feedforward neural networks.