Recovery of a mixture of Gaussians by sum-of-norms clustering

Sum-of-norms clustering is a method for assigning n points in ℝd to K clusters, 1≤ K≤ n, using convex optimization. Recently, Panahi et al. proved that sum-of-norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the number of samples is not too large. The purpose of this note is to lift this restriction, i.e., show that sum-of-norms clustering with equal weights can recover a mixture of Gaussians even as the number of samples tends to infinity. Our proof relies on an interesting characterization of clusters computed by sum-of-norms clustering that was developed inside a proof of the agglomeration conjecture by Chiquet et al. Because we believe this theorem has independent interest, we restate and reprove the Chiquet et al. result herein.