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Submanifold density estimation
Arkadas Ozakin · Alexander Gray

Wed Dec 09 07:00 PM -- 11:59 PM (PST) @

Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much lower-dimensional intrinsic structure. We propose a small modification to kernel density estimation for estimating probability density functions on Riemannian submanifolds of Euclidean space. Using ideas from Riemannian geometry, we prove the consistency of this modified estimator and show that the convergence rate is determined by the intrinsic dimension of the submanifold. We conclude with empirical results demonstrating the behavior predicted by our theory.

Author Information

Arkadas Ozakin (Georgia Institute of Technology)
Alexander Gray (Skytree Inc. and Georgia Tech)

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