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A Note On Spectral Clustering

MPG-Autoren
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Kolev,  Pavel
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

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Mehlhorn,  Kurt
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

Externe Ressourcen
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Volltexte (frei zugänglich)

arXiv:1509.09188.pdf
(Preprint), 251KB

1509.09188v3.pdf
(Preprint), 353KB

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Zitation

Kolev, P., & Mehlhorn, K. (2015). A Note On Spectral Clustering. Retrieved from http://arxiv.org/abs/1509.09188.


Zitierlink: http://hdl.handle.net/11858/00-001M-0000-0028-8F4F-A
Zusammenfassung
Let $G=(V,E)$ be an undirected graph, $\lambda_k$ the $k$th smallest eigenvalue of the normalized Laplacian matrix of $G$, and $\rho(k)$ the smallest value of the maximal conductance over all $k$-way partitions $S_1,\dots,S_k$ of $V$. Peng et al. [PSZ15] gave the first rigorous analysis of $k$-clustering algorithms that use spectral embedding and $k$-means clustering algorithms to partition the vertices of a graph $G$ into $k$ disjoint subsets. Their analysis builds upon a gap parameter $\Upsilon=\rho(k)/\lambda_{k+1}$ that was introduced by Oveis Gharan and Trevisan [GT14]. In their analysis Peng et al. [PSZ15] assume a gap assumption $\Upsilon\geq\Omega(\mathrm{APR}\cdot k^3)$, where $\mathrm{APR}>1$ is the approximation ratio of a $k$-means clustering algorithm. We exhibit an error in one of their Lemmas and provide a correction. With the correction the proof by Peng et al. [PSZ15] requires a stronger gap assumption $\Upsilon\geq\Omega(\mathrm{APR}\cdot k^4)$. Our main contribution is to improve the analysis in [PSZ15] by an $O(k)$ factor. We demonstrate that a gap assumption $\Psi\geq \Omega(\mathrm{APR}\cdot k^3)$ suffices, where $\Psi=\rho_{avr}(k)/\lambda_{k+1}$ and $\rho_{avr}(k)$ is the value of the average conductance of a partition $S_1,\dots,S_k$ of $V$ that yields $\rho(k)$.