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.

在本文中，我们考虑 Bregman k均值问题 (BKM)，它是经典k均值问题的变体。对于n点集\({\mathcal {S}}\)和\(k \le n\)与\(\mu \)相似的 Bregman 散度，BKM 问题的目标首先是找到一个中心子集\(C \subseteq {\mathcal {S}}\)和\( \mid C \mid = k\)然后根据C将\({\mathcal {S}}\)分成k 个簇，使得\(\mu \) 的总和 -来自\({\mathcal {S}}\) 中每个点的类似 Bregman 散度到它最近的中心被最小化。我们通过使用局部搜索方案提出了一个\(\mu \) -类似的 BregMeans++ 算法，并证明该算法值得一个恒定的近似保证。此外，我们扩展了我们的算法来解决称为噪声\(\mu \) -类似 Bregman k -means++ (noisy \(\mu \) -BKM++) 的 BKM 变体，它是噪声场景中的 BKM。出于与 BKM 相同的实例和目的，我们考虑以\(1-\varepsilon _1\)和\(1+ \varepsilon _2\)之间的因子对\(\varepsilon _1 \in [0,1)\)和\(\varepsilon _2 \ge 0\)，并在期望中获得\(O(\log ^2 k)\)的近似比率。