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

TL;DR

This paper develops methods for function recovery from samples with nonuniform evaluation costs, proposing optimal sampling strategies that minimize expected total cost while ensuring accurate, stable reconstruction in polynomial spaces.

Contribution

It introduces a general framework for selecting sampling measures under nonuniform costs, with proven optimal strategies for polynomial spaces with algebraic cost growth.

Findings

Recovery guarantees depend on Christoffel function and Remez constant.

Proposes two optimal sampling strategies for polynomial spaces with algebraic cost functions.

Strategies are proven optimal up to constants and logarithmic factors.

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.