{"title":"A method for searching for a globally optimal k-partition of higher-dimensional datasets","authors":"","doi":"10.1007/s10898-024-01372-6","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>The problem of finding a globally optimal <em>k</em>-partition of a set <span> <span>\\(\\mathcal {A}\\)</span> </span> is a very intricate optimization problem for which in general, except in the case of one-dimensional data, i.e., for data with one feature (<span> <span>\\(\\mathcal {A}\\subset \\mathbb {R}\\)</span> </span>), there is no method to solve. Only in the one-dimensional case, there are efficient methods based on the fact that the search for a globally optimal <em>k</em>-partition is equivalent to solving a global optimization problem for a symmetric Lipschitz-continuous function using the global optimization algorithm <span>DIRECT</span>. In the present paper, we propose a method for finding a globally optimal <em>k</em>-partition in the general case (<span> <span>\\(\\mathcal {A}\\subset \\mathbb {R}^n\\)</span> </span>, <span> <span>\\(n\\ge 1\\)</span> </span>), generalizing an idea for solving the Lipschitz global optimization for symmetric functions. To do this, we propose a method that combines a global optimization algorithm with linear constraints and the <em>k</em>-means algorithm. The first of these two algorithms is used only to find a good initial approximation for the <em>k</em>-means algorithm. The method was tested on a number of artificial datasets and on several examples from the UCI Machine Learning Repository, and an application in spectral clustering for linearly non-separable datasets is also demonstrated. Our proposed method proved to be very efficient. </p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10898-024-01372-6","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The problem of finding a globally optimal k-partition of a set \(\mathcal {A}\) is a very intricate optimization problem for which in general, except in the case of one-dimensional data, i.e., for data with one feature (\(\mathcal {A}\subset \mathbb {R}\)), there is no method to solve. Only in the one-dimensional case, there are efficient methods based on the fact that the search for a globally optimal k-partition is equivalent to solving a global optimization problem for a symmetric Lipschitz-continuous function using the global optimization algorithm DIRECT. In the present paper, we propose a method for finding a globally optimal k-partition in the general case (\(\mathcal {A}\subset \mathbb {R}^n\), \(n\ge 1\)), generalizing an idea for solving the Lipschitz global optimization for symmetric functions. To do this, we propose a method that combines a global optimization algorithm with linear constraints and the k-means algorithm. The first of these two algorithms is used only to find a good initial approximation for the k-means algorithm. The method was tested on a number of artificial datasets and on several examples from the UCI Machine Learning Repository, and an application in spectral clustering for linearly non-separable datasets is also demonstrated. Our proposed method proved to be very efficient.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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