Improved local search algorithms for Bregman k-means and its variants

Abstract

In this paper, we consider the Bregman k-means problem (BKM) which is a variant of the classical k-means problem. For an n-point set ${\mathcal {S}}$ and $k \le n$ with respect to $\mu$-similar Bregman divergence, the BKM problem aims first to find a center subset $C \subseteq {\mathcal {S}}$ with $\mid C \mid = k$ and then separate ${\mathcal {S}}$ into k clusters according to C, such that the sum of $\mu$-similar Bregman divergence from each point in ${\mathcal {S}}$ to its nearest center is minimized. We propose a $\mu$-similar BregMeans++ algorithm by employing the local search scheme, and prove that the algorithm deserves a constant approximation guarantee. Moreover, we extend our algorithm to solve a variant of BKM called noisy $\mu$-similar Bregman k-means++ (noisy $\mu$-BKM++) which is BKM in the noisy scenario. For the same instance and purpose as BKM, we consider the case of sampling a point with an imprecise probability by a factor between $1-\varepsilon _1$ and $1+ \varepsilon _2$ for $\varepsilon _1 \in [0,1)$ and $\varepsilon _2 \ge 0$, and obtain an approximation ratio of $O(\log ^2 k)$ in expectation.