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Convex clustering is a convex relaxation of the $k$-means and hierarchicalclustering. It involves solving a convex optimization problem with theobjective function being a squared error loss plus a fusion penalty thatencourages the estimated centroids for observations in the same cluster to beidentical. However, when data are contaminated, convex clustering with asquared error loss fails even when there is only one arbitrary outlier. Toaddress this challenge, we propose a resistant convex clustering method.Theoretically, we show that the new estimator is resistant to arbitraryoutliers: it does not break down until more than half of the observations arearbitrary outliers. Perhaps surprisingly, the fusion penalty can help enhanceresistance by fusing the estimators to the cluster centers of uncontaminatedsamples, but not the other way around. Numerical studies demonstrate thecompetitive performance of the proposed method.

Resistant convex clustering: How does the fusion penalty enhance resistantance?