Generalized fuzzy c-means (GFCM) is an extension of fuzzy c-means using \(L_{p}\)-norm distances. However, existing methods cannot solve GFCM with m = 1. To solve this problem, we define a new kind of clustering models, called \(L_{p}\)-norm probabilistic K-means (\(L_{p}\)-PKM). Theoretically, \(L_{p}\)-PKM is equivalent to GFCM at m = 1, and can have nonlinear programming solutions based on an efficient active gradient projection (AGP) method, namely, inverse recursion maximum-step active gradient projection (IRMSAGP). On synthetic and UCI datasets, experimental results show that \(L_{p}\)-PKM performs better than GFCM (m > 1) in terms of initialization robustness, p-influence, and clustering performance, and the proposed IRMSAGP also achieves better performance than the traditional AGP in terms of convergence speed.

广义模糊c均值（GFCM）是使用\（L_ {p} \）-范数距离的模糊c均值的扩展。但是，现有方法无法解决m = 1的GFCM 。为解决此问题，我们定义了一种新的聚类模型，称为\（L_ {p} \）-范数概率K均值（\（L_ {p} \） -PKM）。从理论上讲，\（L_ {p} \）- PKM在m = 1时等效于GFCM ，并且可以具有基于有效主动梯度投影（AGP）方法的非线性规划解决方案，即逆递归最大步长主动梯度投影（ IRMSAGP）。在合成数据和UCI数据集上，实验结果表明\（L_ {p} \）- PKM的性能优于GFCM（m> 1）在初始化鲁棒性，p影响和聚类性能方面，所提出的IRMSAGP在收敛速度方面也比传统AGP更好。