Content:
We prove the continuity of maximum-entropy basis functions using Variational Analysis techniques. The use of information-theoretic variational principles to derive basis functions is a recent development. In this setting, data approximation is viewed as an inductive inference problem, with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints. For a set of distinct nodes, the convex approximation of a function u(.) is a linear combination of (non-negative) basis functions pi(.) that reproduces certain affine functions. Given these constraints, we compute pi(.) by minimizing the relative entropy functional (Kullback-Leibler distance). To prove the continuity of the basis functions, we appeal to the theory of epi-convergence.
July 2006