Timezone: »
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)
More from the Same Authors
-
2013 Poster: Which Space Partitioning Tree to Use for Search? »
Parikshit Ram · Alexander Gray -
2012 Poster: Minimax Multi-Task Learning and a Generalized Loss-Compositional Paradigm for MTL »
Nishant A Mehta · Dongryeol Lee · Alexander Gray -
2009 Workshop: Large-Scale Machine Learning: Parallelism and Massive Datasets »
Alexander Gray · Arthur Gretton · Alexander Smola · Joseph E Gonzalez · Carlos Guestrin -
2009 Poster: Linear-time Algorithms for Pairwise Statistical Problems »
Parikshit Ram · Dongryeol Lee · William B March · Alexander Gray -
2009 Spotlight: Linear-time Algorithms for Pairwise Statistical Problems »
Parikshit Ram · Dongryeol Lee · William B March · Alexander Gray -
2009 Poster: Rank-Approximate Nearest Neighbor Search: Retaining Meaning and Speed in High Dimensions »
Parikshit Ram · Dongryeol Lee · Hua Ouyang · Alexander Gray -
2008 Poster: QUIC-SVD: Fast SVD Using Cosine Trees »
Michael Holmes · Alexander Gray · Charles Isbell -
2008 Demonstration: MLPACK: Scalable Machine Learning Software »
Alexander Gray -
2008 Poster: Fast High-dimensional Kernel Summations Using the Monte Carlo Multipole Method »
Dongryeol Lee · Alexander Gray -
2007 Poster: Multi-Stage Monte Carlo Approximation for Fast Generalized Data Summations »
Michael Holmes · Alexander Gray · Charles Isbell