We propose a clustering method that involves chaining four known techniques into a pipeline yielding an algorithm with stronger recovery guarantees than any of the four components separately. Given $n$ points in $\mathbb R^d$, the first component of our pipeline, which we call leapfrog distances, is reminiscent of density-based clustering, yielding an $n\times n$ distance matrix. The leapfrog distances are then translated to new embeddings using multidimensional scaling and spectral methods, two other known techniques, yielding new embeddings of the $n$ points in $\mathbb R^{d'}$, where $d'$ satisfies $d'\ll d$ in general. Finally, sum-of-norms (SON) clustering is applied to the re-embedded points. Although the fourth step (SON clustering) can in principle be replaced by any other clustering method, our focus is on provable guarantees of recovery of underlying structure. Therefore, we establish that the re-embedding improves recovery SON clustering, since SON clustering is a well-studied method that already has provable guarantees.

翻译：我们提出了一种聚类方法，该方法将四种已知技术串联成一条管道，产生一种比四种单独成分具有更强恢复保证的算法。给定 $\mathbb R^d$ 中的 $n$ 个点，我们管道的第一个成分（称为蛙跳距离）类似于基于密度的聚类，产生一个 $n\times n$ 距离矩阵。然后，利用多维缩放和谱方法（另外两种已知技术）将蛙跳距离转换为新的嵌入，得到 $\mathbb R^{d'}$ 中 $n$ 个点的新嵌入，其中 $d'$ 通常满足 $d'\ll d$。最后，对重新嵌入的点应用范数和（SON）聚类。尽管第四步（SON聚类）原则上可以用任何其他聚类方法替代，但我们的重点在于对底层结构恢复的可证明保证。因此，我们证明重新嵌入改进了SON聚类的恢复性能，因为SON聚类是一种已经具有可证明保证的经过充分研究的方法。