Density Level Set Estimation on Manifolds with DBSCAN
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We show that DBSCAN can estimate the connected components of the λ-density level set { x : f(x) >λ} given n i.i.d. samples from an unknown density f. We characterize the regularity of the level set boundaries using parameter β > 0 and analyze the estimation error under the Hausdorff metric. When the data lies in R^D we obtain a rate of O(n^-1/(2β + D)), which matches known lower bounds up to logarithmic factors. When the data lies on an embedded unknown d-dimensional manifold in R^D, then we obtain a rate of O(n^-1/(2β + d·{1, β})). Finally, we provide adaptive parameter tuning in order to attain these rates with no a priori knowledge of the intrinsic dimension, density, or β.
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