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 {x}i=1 in Rd, the convex approximation of a function u(x) is uh(x) = ∑n i=1 pi(x)ui, where {pi}i=1 are nonnegative basis functions, and uh(x) must reproduce affine functions ∑n i=1 pi(x) = 1, ∑n i=1 pi(x)x i = x. Given these constraints, we compute pi(x) by minimizing the relative entropy functional (Kullback–Leibler distance), D(p‖m) = ∑n i=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.