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Abstract Clustering has long been a popular unsupervised learning approach to identify groups of similar objects and discover patterns from unlabeled data in many applications. Yet, coming up with meaningful interpretations of the estimated clusters has often been challenging precisely due to their unsupervised nature. Meanwhile, in many real-world scenarios, there are some noisy supervising auxiliary variables, for instance, subjective diagnostic opinions, that are related to the observed heterogeneity of the unlabeled data. By leveraging information from both supervising auxiliary variables and unlabeled data, we seek to uncover more scientifically interpretable group structures that may be hidden by completely unsupervised analyses. In this work, we propose and develop a new statistical pattern discovery method named supervised convex clustering (SCC) that borrows strength from both information sources and guides towards finding more interpretable patterns via a joint convex fusion penalty. We develop several extensions of SCC to integrate different types of supervising auxiliary variables, to adjust for additional covariates, and to find biclusters. We demonstrate the practical advantages of SCC through simulations and a case study on Alzheimer's disease genomics. Specifically, we discover new candidate genes as well as new subtypes of Alzheimer's disease that can potentially lead to better understanding of the underlying genetic mechanisms responsible for the observed heterogeneity of cognitive decline in older adults. 

Ridge-like regularization often leads to improved generalization performance of machine learning models by mitigating overfitting. While ridge-regularized machine learning methods are widely used in many important applications, direct training via optimization could become challenging in huge data scenarios with millions of examples and features. We tackle such challenges by proposing a general approach that achieves ridge-like regularization through implicit techniques named Minipatch Ridge (MPRidge). Our approach is based on taking an ensemble of coefficients of unregularized learners trained on many tiny, random subsamples of both the examples and features of the training data, which we call minipatches. We empirically demonstrate that MPRidge induces an implicit ridge-like regularizing effect and performs nearly the same as explicit ridge regularization for a general class of predictors including logistic regression, SVM, and robust regression. Embarrassingly parallelizable, MPRidge provides a computationally appealing alternative to inducing ridge-like regularization for improving generalization performance in challenging big-data settings.