Coresets for kernel clustering

doi:10.1007/s10994-024-06540-z

Coresets for kernel clustering

2024-04-22

0

PRE

AI

S

Shaofeng H.-C. Jiang *

DOI：10.1007/s10994-024-06540-z

原文链接

原文求助

摘要

En 中文

We devise coresets for kernel k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Means with a general kernel, and use them to obtain new, more efficient, algorithms. Kernel k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Means has superior clustering capability compared to classical k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Means, particularly when clusters are non-linearly separable, but it also introduces significant computational challenges. We address this computational issue by constructing a coreset, which is a reduced dataset that accurately preserves the clustering costs. Our main result is a coreset for kernel k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Means that works for a general kernel and has size poly(k & varepsilon;-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{poly}\,}}(k\epsilon <^>{-1})$$\end{document}. Our new coreset both generalizes and greatly improves all previous results; moreover, it can be constructed in time near-linear in n. This result immediately implies new algorithms for kernel k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-Means, such as a (1+& varepsilon;)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+\epsilon )$$\end{document}-approximation in time near-linear in n, and a streaming algorithm using space and update time poly(k & varepsilon;-1logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\textrm{poly}\,}}(k \epsilon <^>{-1} \log n)$$\end{document}. We validate our coreset on various datasets with different kernels. Our coreset performs consistently well, achieving small errors while using very few points. We show that our coresets can speed up kernel K-MEANS++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {k-Means++}$$\end{document} (the kernelized version of the widely used K-MEANS++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {k-Means++}$$\end{document} algorithm), and we further use this faster kernel K-MEANS++\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {k-Means++}$$\end{document} for spectral clustering. In both applications, we achieve significant speedup and a better asymptotic growth while the error is comparable to baselines that do not use coresets.

Keyword:

Clustering

Kernel method

Coreset

PTAS

Coresets for kernel clustering

摘要

期刊

机构

引用论文