Sets Clustering

• URL: http://arxiv.org/abs/2003.04135v1
• Date: Mon, 9 Mar 2020 13:30:30 GMT
• Title: Sets Clustering
• Authors: Ibrahim Jubran and Murad Tukan and Alaa Maalouf and Dan Feldman
• Abstract summary: We prove that a core-set of $O(logn)$ sets always exists, and can be computed in $O(nlogn)$ time. Applying an inefficient but optimal algorithm on this coreset allows us to obtain the first PTAS ($1+varepsilon$ approximation) for the sets-$k$-means problem. Open source code and experimental results for document classification and facility locations are also provided.
• Score: 25.358415142404752
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The input to the \emph{sets-$k$-means} problem is an integer $k\geq 1$ and a set $\mathcal{P}=\{P_1,\cdots,P_n\}$ of sets in $\mathbb{R}^d$. The goal is to compute a set $C$ of $k$ centers (points) in $\mathbb{R}^d$ that minimizes the sum $\sum_{P\in \mathcal{P}} \min_{p\in P, c\in C}\left\| p-c \right\|^2$ of squared distances to these sets. An \emph{$\varepsilon$-core-set} for this problem is a weighted subset of $\mathcal{P}$ that approximates this sum up to $1\pm\varepsilon$ factor, for \emph{every} set $C$ of $k$ centers in $\mathbb{R}^d$. We prove that such a core-set of $O(\log^2{n})$ sets always exists, and can be computed in $O(n\log{n})$ time, for every input $\mathcal{P}$ and every fixed $d,k\geq 1$ and $\varepsilon \in (0,1)$. The result easily generalized for any metric space, distances to the power of $z>0$, and M-estimators that handle outliers. Applying an inefficient but optimal algorithm on this coreset allows us to obtain the first PTAS ($1+\varepsilon$ approximation) for the sets-$k$-means problem that takes time near linear in $n$. This is the first result even for sets-mean on the plane ($k=1$, $d=2$). Open source code and experimental results for document classification and facility locations are also provided.

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