Density Estimation Density estimation is arguably the most fundamental statistical task, as all others can be defined in terms of densities. fastlab Big Ideas People Stuff Submanifold Density Estimation Arkadas Ozakin and Alexander Gray Neural Information Processing Systems (NIPS) 2009, appeared 2010 Worst-case theoretical results have led to the common wisdom that kernel density estimation is ineffective in more than a few dimensions. We show that for realistic high-dimensional data, which due to correlations between variables can be modeled as lying on a much lower-dimensional submanifold embedded in the original space, a corrected kernel density estimate is in fact subject to the manifold's dimension. Thus, we show that kernel estimation can be much more effective in high-dimensional settings than previously thought. [pdf] Abstract: 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. @inproceedings{ozakin2010subkde, Author = "Arkadas Ozakin and Alexander G. Gray", Title = "{Submanifold Density Estimation}", booktitle = "Advances in Neural Information Processing Systems (NIPS) 22 (Dec 2009)", Year = "2010", publisher = {MIT Press} } Isee elsewhere Fast Kernel Density Estimation We have developed several algorithmic methods for fast kernel summation, with kernel density estimation as the main example. [see webpage here] Iin progress Hyperkernel-based Density Estimation We have developed a density estimation method based on hyperkernels. QP-based Density Estimation We have developed a density estimation method formulated as a quadratic programming problem.