Optimal Rates for Cluster Tree Estimation using Kernel Density Estimators

We study the problem of optimal estimation of the density cluster tree under various assumptions on the underlying density. Following up on the seminal work of Chaudhuri et al, we formulate a new notion of clustering consistency which is better suited to smooth densites, and derive minimax rates of consistency for cluster tree estimation for H\"{o}lder smooth density of arbitary degree $\alpha$. Our rates depend explicitly in the degree of smoothness of the density and match minimax rates for density estimation under the supremum norm. We exhibit a rate optimal cluster tree estimator, which is derived from a kernel density estimator with an appropaiete choice of the kernel and the bandwidth, assuming knowledge of $\alpha$. We also show that the computationally efficient DBSCAN algorithm achieves the optimal rate for clustering when $\alpha\le 1$ and that a simple variant of the DBSCAN algorithm, is also optimal when the density is Morse and $\alpha =2$. We also derive minimax rates for estimaing the level sets of density with jump discontinuities of prescribed size and demonstrate that a DBSCAN-based algorithm is also rate optimal for this problem.