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Regularized \(k\)-means clustering of high-dimensional data and its asymptotic consistency. (English) Zbl 1335.62109

Summary: \(k\)-means clustering is a widely used tool for cluster analysis due to its conceptual simplicity and computational efficiency. However, its performance can be distorted when clustering high-dimensional data where the number of variables becomes relatively large and many of them may contain no information about the clustering structure. This article proposes a high-dimensional cluster analysis method via regularized \(k\)-means clustering, which can simultaneously cluster similar observations and eliminate redundant variables. The key idea is to formulate the \(k\)-means clustering in a form of regularization, with an adaptive group lasso penalty term on cluster centers. In order to optimally balance the trade-off between the clustering model fitting and sparsity, a selection criterion based on clustering stability is developed. The asymptotic estimation and selection consistency of the regularized \(k\)-means clustering with diverging dimension is established. The effectiveness of the regularized \(k\)-means clustering is also demonstrated through a variety of numerical experiments as well as applications to two gene microarray examples. The regularized clustering framework can also be extended to the general model-based clustering.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
68T10 Pattern recognition, speech recognition
62J07 Ridge regression; shrinkage estimators (Lasso)

Software:

boost; sparcl
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References:

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