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Deriving the continuity of maximumentropy basis functions via variational analysis
Abstract
In this paper, we prove the continuity of maximumentropy basis functions using variational analysis techniques. The use of informationtheoretic 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 Rd, 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 (KullbackLeibler 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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