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 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 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.

中文翻译：

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拟合可处理的凸集以支持函数评估

从其支持函数的评估中估计未知紧致凸集的几何问题出现在一系列科学和工程应用中。传统方法通常依赖于在所有可能的紧凑凸集上最小化误差的估计器；特别是，这些方法允许有限地结合关于基础集合的先验结构信息，并且随着可用测量数量的增加，由此产生的估计变得越来越难以描述。我们通过描述一个框架来解决这两个缺点，该框架用于从支持函数评估中估计易处理的指定凸集。基于凸优化的文献，我们的方法基于估计器，这些估计器将凸集的结构化族的误差最小化，这些凸集被指定为简明描述的集合的线性图像——例如单纯形或谱形——在不比环境空间大得多的高维空间中. 从计算的角度来看，以这种方式参数化的凸集很重要，因为可以有效地优化此类集上的线性函数；在本文的推理上下文中，它们服务于不同的目的，即在重建中加入正则化，同时仍然提供相当大的表达能力。我们提供了我们估计量的渐近行为的几何特征，我们的分析依赖于某些允许半代数描述的集合是 Vapnik-Chervonenkis 类的性质。