轴平行仿射子空间k均值聚类的线性时间近似格式

IF 0.4 4区计算机科学 Q4 MATHEMATICS

Computational Geometry-Theory and Applications Pub Date : 2023-06-01 DOI:10.1016/j.comgeo.2023.101981

Kyungjin Cho, Eunjin Oh

{"title":"轴平行仿射子空间k均值聚类的线性时间近似格式","authors":"Kyungjin Cho, Eunjin Oh","doi":"10.1016/j.comgeo.2023.101981","DOIUrl":null,"url":null,"abstract":"<div>In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with <math><mi>Δ</mi><mo>></mo><mn>0</mn></math> unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an <math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></math>-approximate solution in <math><mi>O</mi><mo>(</mo><mi>n</mi><mi>d</mi><mo>)</mo></math> time. The constants hidden behind <math><mi>O</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math> depend only on <math><mi>Δ</mi><mo>,</mo><mi>ϵ</mi></math> and k. This improves the <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mo>)</mo></math>-time algorithm by Eiben et al. (2021) [7] by a factor of n.</div>","PeriodicalId":51001,"journal":{"name":"Computational Geometry-Theory and Applications","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Linear-time approximation scheme for k-means clustering of axis-parallel affine subspaces\",\"authors\":\"Kyungjin Cho, Eunjin Oh\",\"doi\":\"10.1016/j.comgeo.2023.101981\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with <math><mi>Δ</mi><mo>></mo><mn>0</mn></math> unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an <math><mo>(</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo>)</mo></math>-approximate solution in <math><mi>O</mi><mo>(</mo><mi>n</mi><mi>d</mi><mo>)</mo></math> time. The constants hidden behind <math><mi>O</mi><mo>(</mo><mo>⋅</mo><mo>)</mo></math> depend only on <math><mi>Δ</mi><mo>,</mo><mi>ϵ</mi></math> and k. This improves the <math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>d</mi><mo>)</mo></math>-time algorithm by Eiben et al. (2021) [7] by a factor of n.</div>\",\"PeriodicalId\":51001,\"journal\":{\"name\":\"Computational Geometry-Theory and Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Geometry-Theory and Applications\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0925772123000019\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Geometry-Theory and Applications","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0925772123000019","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

本文给出了d维欧氏空间中不完全数据点的k均值聚类的线性时间近似方案。Δ>；0个未指定条目表示为维度为Δ的轴平行仿射子空间。两个不完全数据点之间的距离被定义为与数据点相对应的轴平行仿射子空间中的两个最近点之间的欧几里得距离。我们提出了一种对维度为Δ的n轴平行仿射子空间进行k均值聚类的算法，该算法在O（nd）时间内产生（1+）-近似解。隐藏在O（‧）后面的常数仅取决于Δ、Ş和k。这改进了Eiben等人的O（n2d）-时间算法。（2021）[7]的因子为n。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Linear-time approximation scheme for k-means clustering of axis-parallel affine subspaces

In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with $Δ > 0$ unspecified entries is represented as an axis-parallel affine subspace of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of n axis-parallel affine subspaces of dimension Δ that yields an $(1 + ϵ)$ -approximate solution in $O (n d)$ time. The constants hidden behind $O (\cdot)$ depend only on $Δ, ϵ$ and k. This improves the $O (n^{2} d)$ -time algorithm by Eiben et al. (2021) [7] by a factor of n.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Computational Geometry-Theory and Applications 数学-数学

CiteScore

1.60

自引率

16.70%

发文量

审稿时长

>12 weeks

期刊介绍： Computational Geometry is a forum for research in theoretical and applied aspects of computational geometry. The journal publishes fundamental research in all areas of the subject, as well as disseminating information on the applications, techniques, and use of computational geometry. Computational Geometry publishes articles on the design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects. Also welcomed are computational geometry solutions to fundamental problems arising in computer graphics, pattern recognition, robotics, image processing, CAD-CAM, VLSI design and geographical information systems. Computational Geometry features a special section containing open problems and concise reports on implementations of computational geometry tools.