This paper deals with unsupervised clustering in high dimensional space. The problem is to estimate both labels and a sparse projection matrix of weights. To address this combinatorial non-convex problem maintaining a strict control on the sparsity of the matrix of weights, we propose an alternating minimization of the Frobenius norm criterion. We provide a new efficient algorithm named k-sparse which alternates k-means with projection-gradient minimization. The projection-gradient step is a method of splitting type, with exact projection on the ℓ1ball to promote sparsity. The convergence of the gradient-projection step is addressed, and a preliminary analysis of the alternating minimization is provided. Experiments on Single Cell RNA sequencing datasets show that our method significantly improves the results of spectral clustering, SIMLR, and Sparcl methods. The complexity of our method is linear with the number of samples (cells), so that the method scales up to large datasets.

This paper aims to build a supervised classifier for dealing with imbalanced datasets, uncertain class proportions, dependencies between features, the presence of both numeric and categorical features, and arbitrary loss functions. The Bayes classifier suffers when prior probability shifts occur between the training and testing sets. A solution is to look for an equalizer decision rule whose class-conditional risks are equal. Such a classifier corresponds to a minimax classifier when it maximizes the Bayes risk. We develop a novel box-constrained minimax classifier which takes into account some constraints on the priors to control the risk maximization. We analyze the empirical Bayes risk with respect to the box-constrained priors for discrete inputs. We show that this risk is a concave non-differentiable multivariate piecewise affine function. A projected subgradient algorithm is derived to maximize this empirical Bayes risk over the box-constrained simplex. Its convergence is established and its speed is bounded. The optimization algorithm is scalable when the number of classes is large. The robustness of our classifier is studied on diverse databases. Our classifier, jointly applied with a clustering algorithm to process mixed attributes, tends to equalize the class-conditional risks while being not too pessimistic.