Fusion rule estimation using vector space methods

In a system of N sensors, the sensor S_j, j = 1, 2..., N, outputs Y^(j) ∈ ℛ, according to an unknown probability distribution P_(Y(j)|X), corresponding to input X ∈ [0, 1]. A training n-sample (X₁, Y₁), (X₂, Y₂),..., (X_n, Y_n) is given where Y_i = (Y _i⁽¹⁾, Y_i⁽²⁾,..., Y_i ^(N)) such that Y_i^(j) is the output of S _j in response to input X_i. The problem is to estimate a fusion rule f : ℛ^N → [0, 1], based on the sample, such that the expected square error is minimized over a family of functions ℱ that constitute a vector space. The function f* that minimizes the expected error cannot be computed since the underlying densities are unknown, and only an approximation f̂ to f* is feasible. We estimate the sample size sufficient to ensure that f̂ provides a close approximation to f* with a high probability. The advantages of vector space methods are two-fold: (1) the sample size estimate is a simple function of the dimensionality of ℱ, and (2) the estimate f̂ can be easily computed by well-known least square methods in polynomial time. The results are applicable to the classical potential function methods and also (to a recently proposed) special class of sigmoidal feedforward neural networks.