Partial recovery bounds for clustering with the relaxed $K$-means

Abstract

We investigate the clustering performances of the relaxed $K$-means in the setting of sub-Gaussian Mixture Model (sGMM) and Stochastic Block Model (SBM). After identifying the appropriate signal-to-noise ratio (SNR), we prove that the misclassification error decays exponentially fast with respect to this SNR. These partial recovery bounds for the relaxed $K$-means improve upon results currently known in the sGMM setting. In the SBM setting, applying the relaxed $K$-means SDP allows us to handle general connection probabilities whereas other SDPs investigated in the literature are restricted to the (dis-)assortative case (where within group probabilities are larger than between group probabilities). Again, this partial recovery bound complements the state-of-the-art results. All together, these results put forward the versatility of the relaxed $K$-means.