DERIVING THE CONTINUITY OF MAXIMUM-ENTROPY BASIS FUNCTIONS VIA VARIATIONAL ANALYSIS.

In this paper, 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 {xi}in=1 in ℝd, the convex approximation of a function u(x) is uh(x) = ∑ in=1 pi(x)ui, where {pi}in=1 are nonnegative basis functions, and uh(x) must reproduce affine functions ∑ in=1 pi(x)= 1, ∑ in=1 pi(x)xi = x. Given these constraints, we compute pi(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p||m) = ∑in=1 pi(x) ln(pi(x)/mi(x)), where mi(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epiconvergence. [ABSTRACT FROM AUTHOR]