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arXiv · 1811.03195

Performance of Johnson-Lindenstrauss Transform for k-Means and k-Medians Clustering

Konstantin Makarychev·Yury Makarychev·Ilya Razenshteyn
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Abstract

Consider an instance of Euclidean $k$-means or $k$-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of $(1+\varepsilon)$ under a projection onto a random $O(\log(k / \varepsilon) / \varepsilon^2)$-dimensional subspace. Further, the cost of every clustering is preserved within $(1+\varepsilon)$. More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean $k$-clustering with the distances raised to the $p$-th power for any constant $p$. For $k$-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for $k$-medians, it answers a question raised by Kannan.

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Konstantin Makarychev, Yury Makarychev, Ilya Razenshteyn. 2018-11-08. Performance of Johnson-Lindenstrauss Transform for k-Means and k-Medians Clustering. https://arxiv.org/abs/1811.03195

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