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.
翻译:北纬组群是一种使用convex优化法将美元点数分配为$mathbb{R ⁇ d$至$K$集群的方法,即1\le K\len n$, 以1\le K\le n$为单位分配。 最近, Panahi 等人\ 证明, 北纬组群总和可以保证回收高斯人混合体, 限制样品数量并不太大。 本说明的目的是取消这一限制, 即显示即使样品数量往往不尽相同, 北纬组群的总和可以回收高斯人混合体。 我们的证据依赖于Chiqueet 等人在证明聚聚点组合内开发的以北纬组群群群群群群组成的有趣的特征描述。 因为我们相信这个标群具有独立的利益, 我们在此重复并重新验证Chiquet et al. 的结果 。