Fitting Tractable Convex Sets to Support Function Evaluations

The geometric problem of estimating an unknown compact convex set from evaluations its support function arises in a range scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the error over all possible sets; particular, these methods allow for limited incorporation prior structural information about underlying resulting estimates become increasingly more complicated to describe as number measurements available grows. We address both shortcomings by describing framework tractably specified sets evaluations. Building literature optimization, our approach is based structured families are linear images concisely described sets—such simplex or spectraplex—in higher-dimensional space not much larger than ambient space. Convex parametrized this manner significant computational perspective one can optimize functionals such efficiently; they serve different purpose inferential context present paper, namely, incorporating regularization reconstruction while still offering considerable expressive power. provide characterization asymptotic behavior estimators, analysis relies property certain which admit semialgebraic descriptions Vapnik–Chervonenkis classes. Our numerical experiments highlight utility previous settings noisy small well those be reconstructed non-polyhedral.