Invariant Coordinate Selection (ICS) is a multivariate technique that relies on the simultaneous diagonalization of two scatter matrices. It serves various purposes, including its use as a dimension reduction tool prior to clustering or outlier detection. Unlike methods such as Principal Component Analysis, ICS has a theoretical foundation that explains why and when the identified subspace should contain relevant information. These general results have been examined in detail primarily for specific scatter combinations within a two-cluster framework. In this study, we expand these investigations to include more clusters and scatter combinations. The case of three clusters in particular is studied at length. Based on these expanded theoretical insights and supported by numerical studies, we conclude that ICS is indeed suitable for recovering Fisher's discriminant subspace under very general settings and cases of failure seem rare.

翻译：不变坐标选择（ICS）是一种依赖于两个散布矩阵同时对角化的多元统计技术。该方法具有多种用途，包括作为聚类或异常值检测前的降维工具。与主成分分析等方法不同，ICS拥有能够解释为何及何时识别出的子空间应包含相关信息的理论基础。这些一般性结果主要在双聚类框架内的特定散布矩阵组合下得到详细验证。本研究将此类探讨扩展至更多聚类数目及散布矩阵组合的情形，其中三聚类案例得到了详尽分析。基于扩展的理论认识并结合数值研究，我们得出结论：在非常一般的设定下，ICS确实适用于恢复Fisher判别子空间，且失效情形似乎较为罕见。