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To address this challenge, we first formulate the <jats:italic>kernel k-means<\/jats:italic> ++ method, which conveys the efficient center initialization strategy of <jats:italic>k<\/jats:italic>-means++ from Euclidean to kernel space. Building on this, we propose <jats:italic>global kernel k-means<\/jats:italic> ++ (<jats:inline-formula>\n              <jats:tex-math>$$\\text {GK}k\\text {M}$$<\/jats:tex-math>\n            <\/jats:inline-formula>++), a novel clustering algorithm designed to balance clustering error minimization with reduced computational cost. <jats:inline-formula>\n              <jats:tex-math>$$\\text {GK}k\\text {M}$$<\/jats:tex-math>\n            <\/jats:inline-formula>++ extends the well-established global kernel <jats:italic>k<\/jats:italic>-means algorithm by incorporating the stochastic initialization strategy of kernel <jats:italic>k<\/jats:italic>-means++. This approach significantly reduces computational complexity while preserving superior clustering error minimization capabilities akin to traditional global kernel <jats:italic>k<\/jats:italic>-means. The experimental results on synthetic, real, and graph datasets indicate that <jats:inline-formula>\n              <jats:tex-math>$$\\text {GK}k\\text {M}$$<\/jats:tex-math>\n            <\/jats:inline-formula>++ consistently outperforms both kernel <jats:italic>k<\/jats:italic>-means with random initialization and kernel <jats:italic>k<\/jats:italic>-means++, while achieving solutions comparable to those provided by the exhaustive and computational intensive global kernel <jats:italic>k<\/jats:italic>-means method.<\/jats:p>","DOI":"10.1007\/s10044-025-01463-4","type":"journal-article","created":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T14:03:49Z","timestamp":1747922629000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient error minimization in kernel k-means clustering"],"prefix":"10.1007","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1352-2062","authenticated-orcid":false,"given":"Georgios","family":"Vardakas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-8668-4477","authenticated-orcid":false,"given":"Ioannis","family":"Papakostas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3170-5428","authenticated-orcid":false,"given":"Aristidis","family":"Likas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,22]]},"reference":[{"issue":"3","key":"1463_CR1","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1145\/331499.331504","volume":"31","author":"AK Jain","year":"1999","unstructured":"Jain AK, Murty MN, Flynn PJ (1999) Data clustering: a review. 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