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Fitting Tractable Convex Sets to Support Function Evaluations

Soh, Yong Sheng and Chandrasekaran, Venkat (2019) Fitting Tractable Convex Sets to Support Function Evaluations. . (Unpublished)

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

Item Type:Report or Paper (Discussion Paper)
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Additional Information:The authors were supported in part by NSF grants CCF-1350590 and CCF-1637598, by Air Force Office of Scientific Research Grant FA9550-16-1-0210, by a Sloan research fellowship, and an A*STAR (Agency for Science, Technology and Research, Singapore) fellowship. A substantial portion of this work was performed while Y. S. Soh was at the California Institute of Technology.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-16-1-0210
Alfred P. Sloan FoundationUNSPECIFIED
Agency for Science, Technology and Research (A*STAR)UNSPECIFIED
Subject Keywords:constrained shape regression, convex regression, entropy of semialgebraic sets, K-means clustering, simplicial polytopes, stochastic equicontinuity
Record Number:CaltechAUTHORS:20190626-093718719
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:96717
Deposited By: Tony Diaz
Deposited On:26 Jun 2019 16:48
Last Modified:03 Oct 2019 21:24

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