On Euclidean k-Means Clustering with alpha-Center Proximity

Amit Deshpande, Anand Louis, Apoorv Singh
Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, PMLR 89:2087-2095, 2019.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T On Euclidean k-Means Clustering with alpha-Center Proximity %A Amit Deshpande %A Anand Louis %A Apoorv Singh %B Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Masashi Sugiyama %F pmlr-v89-deshpande19a %I PMLR %P 2087--2095 %U https://proceedings.mlr.press/v89/deshpande19a.html %V 89 %X $k$-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.
APA
Deshpande, A., Louis, A. & Singh, A.. (2019). On Euclidean k-Means Clustering with alpha-Center Proximity. Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 89:2087-2095 Available from https://proceedings.mlr.press/v89/deshpande19a.html.

Related Material