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- W2504145283 abstract "This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In the case of $k$-medians and $k$-means, the key results are as follows. $bullet$ When the method considers all data points as candidate centers, then selecting $mathcal{O}(klog(1/varepsilon))$ centers achieves cost at most $2+varepsilon$ times the optimal cost with $k$ centers. $bullet$ Alternatively, the same guarantees hold if each round samples $mathcal{O}(k/varepsilon^5)$ candidate centers proportionally to their cluster cost (as with $texttt{kmeans++}$, but holding centers fixed). $bullet$ In the case of $k$-means, considering an augmented set of $n^{lceil1/varepsilonrceil}$ candidate centers gives $1+varepsilon$ approximation with $mathcal{O}(klog(1/varepsilon))$ centers, the entire algorithm taking $mathcal{O}(dklog(1/varepsilon)n^{1+lceil1/varepsilonrceil})$ time, where $n$ is the number of data points in $mathbb{R}^d$. $bullet$ In the case of Euclidean $k$-medians, generating a candidate set via $n^{mathcal{O}(1/varepsilon^2)}$ executions of stochastic gradient descent with adaptively determined constraint sets will once again give approximation $1+varepsilon$ with $mathcal{O}(klog(1/varepsilon))$ centers in $dklog(1/varepsilon)n^{mathcal{O}(1/varepsilon^2)}$ time. Ancillary results include: guarantees for cluster costs based on powers of metrics; a brief, favorable empirical evaluation against $texttt{kmeans++}$; data-dependent bounds allowing $1+varepsilon$ in the first two bullets above, for example with $k$-medians over finite metric spaces." @default.
- W2504145283 created "2016-08-23" @default.
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- W2504145283 date "2016-07-21" @default.
- W2504145283 modified "2023-09-27" @default.
- W2504145283 title "Greedy bi-criteria approximations for $k$-medians and $k$-means" @default.
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