The $K$-means algorithm remains one of the most widely-used clustering methods due to its simplicity and general utility. The performance of $K$-means depends upon location of minima low in cost function, amongst a potentially vast number of solutions. Here, we use the energy landscape approach to map the change in $K$-means solution space as a result of increasing dataset outliers and show that the cost function surface becomes more funnelled. Kinetic analysis reveals that in all cases the overall funnel is composed of shallow locally-funnelled regions, each of which are separated by areas that do not support any clustering solutions. These shallow regions correspond to different types of clustering solution and their increasing number with outliers leads to longer pathways within the funnel and a reduced correlation between accuracy and cost function. Finally, we propose that the rates obtained from kinetic analysis provide a novel measure of clustering similarity that incorporates information about the paths between them. This measure is robust to outliers and we illustrate the application to datasets containing multiple outliers.

本研究使用能量景观方法探寻$K$-means算法中数据集异常值对其性能的影响，发现其代价函数表面会形成更窄的漏斗形态，每个漏斗之间会有一些不支持聚类的区域，而其中的浅漏斗则对应不同类型的聚类解决方案，而异常值的逐渐增多会导致漏斗内的路径变长以及准确性和成本函数之间的相关性降低。最后，本研究提出了一种新的聚类相似度测量方法，能够忽略异常值的影响，并在多异常值的数据集中进行了应用。

数据离群值的添加对K均值解决方案景观的演变及其分析的鲁棒聚类比较度量