On the power of linear programming for K-means clustering
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908981260189696 |
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| author | De Rosa, Antonio Khajavirad, Aida Wang, Yakun |
| author_facet | De Rosa, Antonio Khajavirad, Aida Wang, Yakun |
| contents | In [SIAM J. Optim., 2022], the authors introduced a new linear programming (LP) relaxation for K-means clustering. In this paper, we further investigate both theoretical and computational properties of this relaxation. As evident from our numerical experiments with both synthetic real-world data sets, the proposed LP relaxation is almost always tight; i.e. its optimal solution is feasible for the original nonconvex problem. To better understand this unexpected behaviour, on the theoretical side, we focus on K-means clustering with two clusters, and we obtain sufficient conditions under which the LP relaxation is tight. We further analyze the sufficient conditions when the input is generated according to a popular stochastic model and obtain recovery guarantees for the LP relaxation. We conclude our theoretical study by constructing a family of inputs for which the LP relaxation is never tight. Denoting by $n$ the number of data points to be clustered, the LP relaxation contains $Ω(n^3)$ inequalities making it impractical for large data sets. To address the scalability issue, by building upon a cutting-plane algorithm together with the GPU implementation of PDLP, a first-order method LP solver, we develop an efficient algorithm that solves the proposed LP and hence the K-means clustering problem, for up to $n \leq 4000$ data points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the power of linear programming for K-means clustering De Rosa, Antonio Khajavirad, Aida Wang, Yakun Optimization and Control 90C05, 90C57, 62H30, 49Q20, 68Q87 In [SIAM J. Optim., 2022], the authors introduced a new linear programming (LP) relaxation for K-means clustering. In this paper, we further investigate both theoretical and computational properties of this relaxation. As evident from our numerical experiments with both synthetic real-world data sets, the proposed LP relaxation is almost always tight; i.e. its optimal solution is feasible for the original nonconvex problem. To better understand this unexpected behaviour, on the theoretical side, we focus on K-means clustering with two clusters, and we obtain sufficient conditions under which the LP relaxation is tight. We further analyze the sufficient conditions when the input is generated according to a popular stochastic model and obtain recovery guarantees for the LP relaxation. We conclude our theoretical study by constructing a family of inputs for which the LP relaxation is never tight. Denoting by $n$ the number of data points to be clustered, the LP relaxation contains $Ω(n^3)$ inequalities making it impractical for large data sets. To address the scalability issue, by building upon a cutting-plane algorithm together with the GPU implementation of PDLP, a first-order method LP solver, we develop an efficient algorithm that solves the proposed LP and hence the K-means clustering problem, for up to $n \leq 4000$ data points. |
| title | On the power of linear programming for K-means clustering |
| topic | Optimization and Control 90C05, 90C57, 62H30, 49Q20, 68Q87 |
| url | https://arxiv.org/abs/2402.01061 |