8 research outputs found
Projection Estimates of Constrained Functional Parameters
AMS classifications: 62G05; 62G07; 62G08; 62G20; 62G32;estimation;convex function;extreme value copula;Pickands dependence function;projection;shape constraint;support function;tangent cone
Fitting Tractable Convex Sets to Support Function Evaluations
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 spectraplex -- 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.Comment: 35 pages, 80 figure
Local tests for consistency of support hyperplane data
Caption title.Includes bibliographical references (p. 32-33).Supported by the U.S. Army Research Office. DAAL03-92-G-0115 DAAL03-92-G-0320 Supported by the Office of Naval Research. N00014-91-J-1004 Supported by the National Science Foundation. MIP-9015281 IRI-9209577William C. Karl ... [et al.]
Projection Estimates of Constrained Functional Parameters
AMS classifications: 62G05; 62G07; 62G08; 62G20; 62G32;
Convex set estimation from support line measurements and applications to target reconstruction from laser radar data
Cover title.Includes bibliographical references (p. 42-44).Research supported by the Department of the Navy under an Air Force contract. F19628-90-C-0002 Research supported by the U.S. Army Research Office. DAAL03-86-K-0171 Research supported by the National Science Foundation. ECS-8700903A.S. Lele, S.R. Kulkarni, A.S. Willsky
Fitting Tractable Convex Sets to Support Function Evaluations
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 allow for limited 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 setsāsuch as the simplex or the spectraplexāin 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 VapnikāChervonenkis 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