Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.10613 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913876189118464 |
|---|---|
| author | Adcock, Ben |
| author_facet | Adcock, Ben |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10613 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Function recovery and optimal sampling in the presence of nonuniform evaluation costs Adcock, Ben Numerical Analysis 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. |
| title | Function recovery and optimal sampling in the presence of nonuniform evaluation costs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2502.10613 |