Recovery of a mixture of Gaussians by sum-of-norms clustering
This provides a theoretical guarantee for clustering algorithms in machine learning, though it appears incremental as it extends prior work by removing a sample size limitation.
The paper tackles the problem of recovering a mixture of Gaussians using sum-of-norms clustering, showing that this method can achieve recovery even as the number of samples tends to infinity, lifting a previous restriction.
Sum-of-norms clustering is a method for assigning $n$ points in $\mathbb{R}^d$ to $K$ clusters, $1\le K\le n$, 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.