clustering step). The work [@BinYu] formulates this problem in a precise mathematical setting, by introducing a notion of a well-separated data generating mixture model. Roughly speaking, the authors obtain a result that, in spirit, reads similarly to Theorem \[thm:discrete-is-consistent-with-continuum\]. However, the notion of a well-separated model introduced in that work depends on extrinsic quantities such as the kernel used to construct the graph at the finite data level, and in particular they work in the regime where $n$ is thought to be large, but the connectivity length-scale ${\varepsilon}$ does not decay to zero. The article [@BinYu] has inspired and motivated our work.

Different scenarios where one can define a notion of well-separated components were studied in [@LittleMaggioniMurphy; @LingStromer]. Both of these papers are closely related to our work, and we believe are complementary of each other. In both works, the respective notion of “well-separated components" is used to help validate proposed methodologies for clustering. In [@LittleMaggioniMurphy], the focus is on understanding the stability of spectral embeddings in situations where the original data set has a geometric structure that is highly anisotropic. The authors propose the use of a general family of distances (other than the Euclidean one) to construct graphs 