High-Dimensional Sparse Clustering via Iterative Semidefinite Programming Relaxed K-Means

We study high-dimensional clustering where the true signal is sparse among features. To help design algorithms that do not rely on precise estimation of sparse nuisance parameters or signal feature set, we establish minimax separation for exact cluster recovery of semidefinite programming (SDP) relaxation of $K$-means using varying subsets of features. It highlights the critical role of feature selection and, at the same time, shows that the signal feature set can be slightly overestimated without compromising clustering performance. Guided by this theory, our method alternates between rough feature selection and clustering: feature selection is performed by thresholding a rough estimate of the discriminative direction, while clustering is carried out via an SDP-relaxed $K$-means. We further extend the method to settings with unknown sparse precision matrices, avoiding full model parameter estimation by computing only the minimally required quantities.