Function recovery and optimal sampling in the presence of nonuniform evaluation costs

Ben Adcock

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arXiv · 2502.10613

Function recovery and optimal sampling in the presence of nonuniform evaluation costs

Ben Adcock

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Abstract

We consider recovering a function $f : D \rightarrow \mathbb{C}$ in an $n$-dimensional linear subspace $\mathcal{P}$ from i.i.d. pointwise samples via (weighted) least-squares estimators. Different from most works, we assume the cost of evaluating $f$ is potentially nonuniform, and governed by a cost function $c : D \rightarrow (0,\infty)$ which may blow up at certain points. We therefore strive to choose the sampling measure in a way that minimizes the expected total cost. We provide a recovery guarantee which asserts accurate and stable recovery with an expected cost depending on the Christoffel function and Remez constant of the space $\mathcal{P}$. This leads to a general recipe for finding a good sampling measure for general $c$. As an example, we consider one-dimensional polynomial spaces. Here, we provide two strategies for choosing the sampling measure, which we prove are optimal (up to constants and log factors) in the case of algebraically-growing cost functions.

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Ben Adcock. 2025-02-15. Function recovery and optimal sampling in the presence of nonuniform evaluation costs. https://arxiv.org/abs/2502.10613

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