  While K-means is known to be a standard clustering algorithm, it may be
compromised due to the presence of outliers and high-dimensional noisy
variables. This paper proposes adaptively robust and sparse K-means clustering
(ARSK) to address these practical limitations of the standard K-means
algorithm. We introduce a redundant error component for each observation for
robustness, and this additional parameter is penalized using a group sparse
penalty. To accommodate the impact of high-dimensional noisy variables, the
objective function is modified by incorporating weights and implementing a
penalty to control the sparsity of the weight vector. The tuning parameters to
control the robustness and sparsity are selected by Gap statistics. Through
simulation experiments and real data analysis, we demonstrate the superiority
of the proposed method to existing algorithms in identifying clusters without
outliers and informative variables simultaneously.
