Fast $k$-means Seeding Under The Manifold Hypothesis
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912866808889344 |
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| author | Shah, Poojan Agrawal, Shashwat Jaiswal, Ragesh |
| author_facet | Shah, Poojan Agrawal, Shashwat Jaiswal, Ragesh |
| contents | We study beyond worst case analysis for the $k$-means problem where the goal is to model typical instances of $k$-means arising in practice. Existing theoretical approaches provide guarantees under certain assumptions on the optimal solutions to $k$-means, making them difficult to validate in practice. We propose the manifold hypothesis, where data obtained in ambient dimension $D$ concentrates around a low dimensional manifold of intrinsic dimension $d$, as a reasonable assumption to model real world clustering instances. We identify key geometric properties of datasets which have theoretically predictable scaling laws depending on the quantization exponent $\varepsilon = 2/d$ using techniques from optimum quantization theory. We show how to exploit these regularities to design a fast seeding method called $\operatorname{Qkmeans}$ which provides $O(ρ^{-2} \log k)$ approximate solutions to the $k$-means problem in time $O(nD) + \widetilde{O}(\varepsilon^{1+ρ}ρ^{-1}k^{1+γ})$; where the exponent $γ= \varepsilon + ρ$ for an input parameter $ρ< 1$. This allows us to obtain new runtime - quality tradeoffs. We perform a large scale empirical study across various domains to validate our theoretical predictions and algorithm performance to bridge theory and practice for beyond worst case data clustering. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_01104 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast $k$-means Seeding Under The Manifold Hypothesis Shah, Poojan Agrawal, Shashwat Jaiswal, Ragesh Data Structures and Algorithms We study beyond worst case analysis for the $k$-means problem where the goal is to model typical instances of $k$-means arising in practice. Existing theoretical approaches provide guarantees under certain assumptions on the optimal solutions to $k$-means, making them difficult to validate in practice. We propose the manifold hypothesis, where data obtained in ambient dimension $D$ concentrates around a low dimensional manifold of intrinsic dimension $d$, as a reasonable assumption to model real world clustering instances. We identify key geometric properties of datasets which have theoretically predictable scaling laws depending on the quantization exponent $\varepsilon = 2/d$ using techniques from optimum quantization theory. We show how to exploit these regularities to design a fast seeding method called $\operatorname{Qkmeans}$ which provides $O(ρ^{-2} \log k)$ approximate solutions to the $k$-means problem in time $O(nD) + \widetilde{O}(\varepsilon^{1+ρ}ρ^{-1}k^{1+γ})$; where the exponent $γ= \varepsilon + ρ$ for an input parameter $ρ< 1$. This allows us to obtain new runtime - quality tradeoffs. We perform a large scale empirical study across various domains to validate our theoretical predictions and algorithm performance to bridge theory and practice for beyond worst case data clustering. |
| title | Fast $k$-means Seeding Under The Manifold Hypothesis |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2602.01104 |