Optimal recovery is a mathematical framework for learning functions from observational data by adopting a worst-case perspective tied to model assumptions on the functions to be learned. Working in a finite-dimensional Hilbert space, we consider model assumptions based on approximability and observation inaccuracies modeled as additive errors bounded in $\ell_2$. We focus on the local recovery problem, which amounts to the determination of Chebyshev centers. Earlier work by Beck and Eldar presented a semidefinite recipe for the determination of Chebyshev centers. The result was valid in the complex setting only, but not necessarily in the real setting, since it relied on the S-procedure with two quadratic constraints, which offers a tight relaxation only in the complex setting. Our contribution consists in proving that this semidefinite recipe is exact in the real setting, too, at least in the particular instance where the quadratic constraints involve orthogonal projectors. Our argument exploits a previous work of ours, where exact Chebyshev centers were obtained in a different way. We conclude by stating some open questions and by commenting on other recent results in optimal recovery.

翻译：最优恢复是一种数学框架，通过采用与待学习函数模型假设相关的悲观视角，从观测数据中学习函数。在有限维希尔伯特空间中，我们考虑基于可逼近性的模型假设，以及以$\ell_2$范数有界加性误差建模的观测不准确性。我们聚焦于局部恢复问题，该问题等价于确定切比雪夫中心。Beck和Eldar的早期工作提出了一种用于确定切比雪夫中心的半定规划方法。该结果仅在复域中有效，但在实域中未必成立，因为它依赖于包含两个二次约束的S-过程，而该过程仅在复域中提供紧松弛。我们的贡献在于证明：至少在二次约束涉及正交投影算子的特定情形下，该半定规划方法在实域中也具有精确性。我们的论证利用了先前的研究工作（其中以不同方式获得了精确切比雪夫中心）。最后，我们提出若干开放性问题，并评述最优恢复领域其他近期研究成果。