Model-based approaches to cluster analysis and mixture modeling often involve maximizing classification and mixture likelihoods. Without appropriate constrains on the scatter matrices of the components, these maximizations result in ill-posed problems. Moreover, without constrains, non-interesting or “spurious” clusters are often detected by the EM and CEM algorithms traditionally used for the maximization of the likelihood criteria. Considering an upper bound on the maximal ratio between the determinants of the scatter matrices seems to be a sensible way to overcome these problems by affine equivariant constraints. Unfortunately, problems still arise without also controlling the elements of the “shape” matrices. A new methodology is proposed that allows both control of the scatter matrices determinants and also the shape matrices elements. Some theoretical justification is given. A fast algorithm is proposed for this doubly constrained maximization. The methodology is also extended to robust model-based clustering problems.

中文翻译：

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行列式约束的基于模型的聚类

基于模型的聚类分析和混合建模方法通常涉及最大化分类和混合可能性。在组件的散布矩阵上没有适当的约束时，这些最大化会导致不适定的问题。此外，在没有限制的情况下，传统上用于最大化似然准则的EM和CEM算法通常会检测到不感兴趣或“虚假”的群集。考虑散点矩阵的行列式之间的最大比率的上限似乎是通过仿射等变约束来克服这些问题的明智方法。不幸的是，如果不同时控制“形状”矩阵的元素，仍然会出现问题。提出了一种新方法，该方法既可以控制散射矩阵行列式，也可以控制形状矩阵元素。给出了一些理论上的证明。针对这种双重约束最大化提出了一种快速算法。该方法还扩展到基于模型的鲁棒聚类问题。