# Optimal Recovery from Inaccurate Data in Hilbert Spaces: Regularize, but what of the Parameter?

arXiv: https://arxiv.org/abs/2111.02601 · HTML: https://mlsanity.com/paper/2111.02601/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 25 references that are in the map.
- First posted on arXiv 4 Nov 2021.

## Builds on (most cited first, 3 of 3)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Instances of Computational Optimal Recovery: Dealing with Observation Errors](https://mlsanity.com/paper/2004.00192/) (2020)

## Built on by (most cited first, 2 of 2)

- [On the Optimal Recovery of Graph Signals](https://mlsanity.com/paper/2304.00474/) (2023)
- [Optimal Recovery Meets Bayesian Learning: Where Worst-Case Bounds Pay Off](https://mlsanity.com/paper/2609.29622/) (2026)

## Abstract (arXiv)

In Optimal Recovery, the task of learning a function from observational data is tackled deterministically by adopting a worst-case perspective tied to an explicit model assumption made on the functions to be learned. Working in the framework of Hilbert spaces, this article considers a model assumption based on approximability. It also incorporates observational inaccuracies modeled via additive errors bounded in ℓ2. Earlier works have demonstrated that regularization provide algorithms that are optimal in this situation, but did not fully identify the desired hyperparameter. This article fills the gap in both a local scenario and a global scenario. In the local scenario, which amounts to the determination of Chebyshev centers, the semidefinite recipe of Beck and Eldar (legitimately valid in the complex setting only) is complemented by a more direct approach, with the proviso that the observational functionals have orthonormal representers. In the said approach, the desired parameter is the solution to an equation that can be resolved via standard methods. In the global scenario, where linear algorithms rule, the parameter elusive in the works of Micchelli et al. is found as the byproduct of a semidefinite program. Additionally and quite surprisingly, in case of observational functionals with orthonormal representers, it is established that any regularization parameter is optimal.