Optimal Coresets for Low-Dimensional Geometric Median

Peyman Afshani, Chris Schwiegelshohn
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:262-270, 2024.

Abstract

We investigate coresets for approximating the cost with respect to median queries. In this problem, we are given a set of points $P\subset \mathbb{R}^d$ and median queries are $\sum_{p\in P} ||p-c||$ for any point $c\in \mathbb{R}^d$. Our goal is to compute a small weighted summary $S\subset P$ such that the cost of any median query is approximated within a multiplicative $(1\pm\varepsilon)$ factor. We provide matching upper and lower bounds on the number of points contained in $S$ of the order $\tilde{\Theta}\left(\varepsilon^{-d/(d+1)}\right)$.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-afshani24a, title = {Optimal Coresets for Low-Dimensional Geometric Median}, author = {Afshani, Peyman and Schwiegelshohn, Chris}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {262--270}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/afshani24a/afshani24a.pdf}, url = {https://proceedings.mlr.press/v235/afshani24a.html}, abstract = {We investigate coresets for approximating the cost with respect to median queries. In this problem, we are given a set of points $P\subset \mathbb{R}^d$ and median queries are $\sum_{p\in P} ||p-c||$ for any point $c\in \mathbb{R}^d$. Our goal is to compute a small weighted summary $S\subset P$ such that the cost of any median query is approximated within a multiplicative $(1\pm\varepsilon)$ factor. We provide matching upper and lower bounds on the number of points contained in $S$ of the order $\tilde{\Theta}\left(\varepsilon^{-d/(d+1)}\right)$.} }
Endnote
%0 Conference Paper %T Optimal Coresets for Low-Dimensional Geometric Median %A Peyman Afshani %A Chris Schwiegelshohn %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-afshani24a %I PMLR %P 262--270 %U https://proceedings.mlr.press/v235/afshani24a.html %V 235 %X We investigate coresets for approximating the cost with respect to median queries. In this problem, we are given a set of points $P\subset \mathbb{R}^d$ and median queries are $\sum_{p\in P} ||p-c||$ for any point $c\in \mathbb{R}^d$. Our goal is to compute a small weighted summary $S\subset P$ such that the cost of any median query is approximated within a multiplicative $(1\pm\varepsilon)$ factor. We provide matching upper and lower bounds on the number of points contained in $S$ of the order $\tilde{\Theta}\left(\varepsilon^{-d/(d+1)}\right)$.
APA
Afshani, P. & Schwiegelshohn, C.. (2024). Optimal Coresets for Low-Dimensional Geometric Median. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:262-270 Available from https://proceedings.mlr.press/v235/afshani24a.html.

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