Query-Efficient Fixpoints of $\ell_p$-Contractions
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913763141091328 |
|---|---|
| author | Haslebacher, Sebastian Lill, Jonas Schnider, Patrick Weber, Simon |
| author_facet | Haslebacher, Sebastian Lill, Jonas Schnider, Patrick Weber, Simon |
| contents | We prove that an $ε$-approximate fixpoint of a map $f:[0,1]^d\rightarrow [0,1]^d$ can be found with $\mathcal{O}(d^2(\log\frac{1}ε + \log\frac{1}{1-λ}))$ queries to $f$ if $f$ is $λ$-contracting with respect to an $\ell_p$-metric for some $p\in [1,\infty)\cup\{\infty\}$. This generalizes a recent result of Chen, Li, and Yannakakis [STOC'24] from the $\ell_\infty$-case to all $\ell_p$-metrics. Previously, all query upper bounds for $p\in [1,\infty) \setminus \{2\}$ were either exponential in $d$, $\log\frac{1}ε$, or $\log\frac{1}{1-λ}$.
Chen, Li, and Yannakakis also show how to ensure that all queries to $f$ lie on a discrete grid of limited granularity in the $\ell_\infty$-case. We provide such a rounding for the $\ell_1$-case, placing an appropriately defined version of the $\ell_1$-case in $\textsf{FP}^{dt}$.
To prove our results, we introduce the notion of $\ell_p$-halfspaces and generalize the classical centerpoint theorem from discrete geometry: for any $p \in [1, \infty) \cup \{\infty\}$ and any mass distribution (or point set), we prove that there exists a centerpoint $c$ such that every $\ell_p$-halfspace defined by $c$ and a normal vector contains at least a $\frac{1}{d+1}$-fraction of the mass (or points). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16089 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Query-Efficient Fixpoints of $\ell_p$-Contractions Haslebacher, Sebastian Lill, Jonas Schnider, Patrick Weber, Simon Computational Complexity Computational Geometry We prove that an $ε$-approximate fixpoint of a map $f:[0,1]^d\rightarrow [0,1]^d$ can be found with $\mathcal{O}(d^2(\log\frac{1}ε + \log\frac{1}{1-λ}))$ queries to $f$ if $f$ is $λ$-contracting with respect to an $\ell_p$-metric for some $p\in [1,\infty)\cup\{\infty\}$. This generalizes a recent result of Chen, Li, and Yannakakis [STOC'24] from the $\ell_\infty$-case to all $\ell_p$-metrics. Previously, all query upper bounds for $p\in [1,\infty) \setminus \{2\}$ were either exponential in $d$, $\log\frac{1}ε$, or $\log\frac{1}{1-λ}$. Chen, Li, and Yannakakis also show how to ensure that all queries to $f$ lie on a discrete grid of limited granularity in the $\ell_\infty$-case. We provide such a rounding for the $\ell_1$-case, placing an appropriately defined version of the $\ell_1$-case in $\textsf{FP}^{dt}$. To prove our results, we introduce the notion of $\ell_p$-halfspaces and generalize the classical centerpoint theorem from discrete geometry: for any $p \in [1, \infty) \cup \{\infty\}$ and any mass distribution (or point set), we prove that there exists a centerpoint $c$ such that every $\ell_p$-halfspace defined by $c$ and a normal vector contains at least a $\frac{1}{d+1}$-fraction of the mass (or points). |
| title | Query-Efficient Fixpoints of $\ell_p$-Contractions |
| topic | Computational Complexity Computational Geometry |
| url | https://arxiv.org/abs/2503.16089 |