Instances of computational optimal recovery: Refined approximability models
Instances of computational optimal recovery: Refined approximability models
复制标题
计算最优恢复实例：精炼的近似模型
DOI：
10.1016/j.jco.2020.101503
复制
发表时间：
2020
影响因子：
1.7
通讯作者：
Foucart, Simon
中科院分区：
文献类型：
--
作者：
Foucart, Simon
摘要 最近在最优恢复的背景下研究了基于近似能力的模型。然而，这些模型与过度参数化不兼容，因为模型和数据一致的函数可能是无界的。这一缺点促使人们引入具有附加边界条件的精炼近似模型。因此，本文提出了两种新模型：一种将有界性应用于目标函数（第一种类型），另一种将有界性应用于近似值（第二种类型）。对于这两种类型的模型，在解决其有效构造之前，首先在抽象级别上描述线性泛函恢复的最佳映射。通过利用半定规划技术，这些构造在涉及 C [− 1, 1] 多项式子空间的常见示例上显式执行。
Abstract Models based on approximation capabilities have recently been studied in the context of Optimal Recovery. These models, however, are not compatible with overparametrization, since model-and data-consistent functions could then be unbounded. This drawback motivates the introduction of refined approximability models featuring an added boundedness condition. Thus, two new models are proposed in this article: one where the boundedness applies to the target functions (first type) and one where the boundedness applies to the approximants (second type). For both types of models, optimal maps for the recovery of linear functionals are first described on an abstract level before their efficient constructions are addressed. By exploiting techniques from semidefinite programming, these constructions are explicitly carried out on a common example involving polynomial subspaces of C [− 1, 1].