{"title":"Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces","authors":"K. Cho, Eunjin Oh","doi":"10.4230/LIPIcs.ISAAC.2021.46","DOIUrl":null,"url":null,"abstract":"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 ∆ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension ∆. 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 axis-parallel affine subspaces of dimension ∆ that yields an (1+ ϵ )-approximate solution in O ( nd ) time. The constants hidden behind O ( · ) depend only on ∆ , ϵ and k . This improves the O ( n 2 d )-time algorithm by Eiben et al. [SODA’21] by a factor of n .","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"1 1","pages":"101981"},"PeriodicalIF":0.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discret. Comput. Geom.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.ISAAC.2021.46","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
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 ∆ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension ∆. 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 axis-parallel affine subspaces of dimension ∆ that yields an (1+ ϵ )-approximate solution in O ( nd ) time. The constants hidden behind O ( · ) depend only on ∆ , ϵ and k . This improves the O ( n 2 d )-time algorithm by Eiben et al. [SODA’21] by a factor of n .