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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)}(i=1)(n) in R-d, the convex approximation of a function u(x) is u(h)(x) = Sigma(n)(i=1) p(i)(x)u(i), where {p(i)}(i=1)(n) are nonnegative basis functions, and u(h)(x) must reproduce a. ne functions Sigma(n)(i=1) p(i)(x) = 1, Sigma(n)(i=1) p(i)(x) x(i) = x. Given these constraints, we compute p(i)(x) by minimizing the relative entropy functional (Kullback-Leibler distance), D(p parallel to m) = Sigma(n)(i=1) p(i)(x) ln(p(i)(x)/m(i)(x)), where m(i)(x) is a known prior weight function distribution. To prove the continuity of the basis functions, we appeal to the theory of epiconvergence.

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