Sets Clustering

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 $\mathbb{R}<sup>d$. The goal is to compute a set $C$ of $k$ centers (points) in $\mathbb{R}<sup>d$ that minimizes the sum $$\sum_{P\in \mathcal{P}} \min_{p\in P, c\in C}\left| p-c \right|<sup>2$$ 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 $\mathbb{R}<sup>d$. We prove that such a core-set of $O(\log<sup>2{n})$ 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 $z&gt;0$, 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.