Topological Equivalence of Spectral and Density Cluster Hierarchies

We compare two canonical ways to recover hierarchical clustering from an underlying density $f$: the density cluster tree given by connected components of superlevel sets $\{f\geλ\}$, and a spectral hierarchy obtained by applying single linkage to diffusion distances induced by a Laplacian operator. We give conditions under which, at an appropriate diffusion time, the diffusion hierarchy is topologically equivalent to the density cluster tree up to a monotone rescaling of heights. It further shows that a $k$-NN graph implementation inherits this equivalence from uniform approximation of diffusion distances, yielding a finite-sample reduction from spectral graph constructions to density-level-set topology.