Density Level Set Estimation on Manifolds with DBSCAN
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Abstract
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 
we obtain a rate of 
which matches known lower bounds up to logarithmic factors. When the data lies on an embedded unknown ddimensional manifold in 
, then we obtain a rate of 
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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