On Euclidean k-Means Clustering with alpha-Center Proximity
Amit Deshpande, Anand Louis, Apoorv Singh
Abstract
$k$-means clustering is NP-hard in the worst case but previous work has shown efficient algorithms assuming the optimal $k$-means clusters are \emph{stable} under additive or multiplicative perturbation of data. This has two caveats. First, we do not know how to efficiently verify this property of optimal solutions that are NP-hard to compute in the first place. Second, the stability assumptions required for polynomial time $k$-means algorithms are often unreasonable when compared to the ground-truth clusters in real-world data. A consequence of multiplicative perturbation resilience is \emph{center proximity}, that is, every point is closer to the center of its own cluster than the center of any other cluster, by some multiplicative factor $\alpha > 1$. We study the problem of minimizing the Euclidean $k$-means objective only over clusterings that satisfy $\alpha$-center proximity. We give a simple algorithm to find the optimal $\alpha$-center-proximal $k$-means clustering in running time exponential in $k$ and $1/(\alpha - 1)$ but linear in the number of points and the dimension. We define an analogous $\alpha$-center proximity condition for outliers, and give similar algorithmic guarantees for $k$-means with outliers and $\alpha$-center proximity. On the hardness side we show that for any $\alpha’ > 1$, there exists an $\alpha \leq \alpha’$, $(\alpha >1)$, and an $\e_0 > 0$ such that minimizing the $k$-means objective over clusterings that satisfy $\alpha$-center proximity is NP-hard to approximate within a multiplicative $(1+\e_0)$ factor.
BibTeX
@InProceedings{pmlr-v89-deshpande19a,
title = {On Euclidean k-Means Clustering with alpha-Center Proximity},
author = {Deshpande, Amit and Louis, Anand and Singh, Apoorv},
booktitle = {Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics},
pages = {2087--2095},
year = {2019},
editor = {Chaudhuri, Kamalika and Sugiyama, Masashi},
volume = {89},
series = {Proceedings of Machine Learning Research},
month = {16--18 Apr},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v89/deshpande19a/deshpande19a.pdf},
url = {https://proceedings.mlr.press/v89/deshpande19a.html},
abstract = {$k$-means clustering is NP-hard in the worst case but previous work has shown efficient algorithms assuming the optimal $k$-means clusters are \emph{stable} under additive or multiplicative perturbation of data. This has two caveats. First, we do not know how to efficiently verify this property of optimal solutions that are NP-hard to compute in the first place. Second, the stability assumptions required for polynomial time $k$-means algorithms are often unreasonable when compared to the ground-truth clusters in real-world data. A consequence of multiplicative perturbation resilience is \emph{center proximity}, that is, every point is closer to the center of its own cluster than the center of any other cluster, by some multiplicative factor $\alpha > 1$. We study the problem of minimizing the Euclidean $k$-means objective only over clusterings that satisfy $\alpha$-center proximity. We give a simple algorithm to find the optimal $\alpha$-center-proximal $k$-means clustering in running time exponential in $k$ and $1/(\alpha - 1)$ but linear in the number of points and the dimension. We define an analogous $\alpha$-center proximity condition for outliers, and give similar algorithmic guarantees for $k$-means with outliers and $\alpha$-center proximity. On the hardness side we show that for any $\alpha’ > 1$, there exists an $\alpha \leq \alpha’$, $(\alpha >1)$, and an $\e_0 > 0$ such that minimizing the $k$-means objective over clusterings that satisfy $\alpha$-center proximity is NP-hard to approximate within a multiplicative $(1+\e_0)$ factor.}
}