Dongryeol Lee, Andrew Moore, Alexander Gray
In previous work we presented an efﬁcient approach to computing ker- nel summations which arise in many machine learning methods such as kernel density estimation. This approach, dual-tree recursion with ﬁnite- difference approximation, generalized existing methods for similar prob- lems arising in computational physics in two ways appropriate for sta- tistical problems: toward distribution sensitivity and general dimension, partly by avoiding series expansions. While this proved to be the fastest practical method for multivariate kernel density estimation at the optimal bandwidth, it is much less efﬁcient at larger-than-optimal bandwidths. In this work, we explore the extent to which the dual-tree approach can be integrated with multipole-like Hermite expansions in order to achieve reasonable efﬁciency across all bandwidth scales, though only for low di- mensionalities. In the process, we derive and demonstrate the ﬁrst truly hierarchical fast Gauss transforms, effectively combining the best tools from discrete algorithms and continuous approximation theory.
1 Fast Gaussian Summation
Kernel summations are fundamental in both statistics/learning and computational physics.