Faster Gaussian Summation: Theory and Experiment

We provide faster algorithms for the prob- lem of Gaussian summation, which occurs in many machine learning methods. We de- velop two new extensions - an O(Dp) Tay- lor expansion for the Gaussian kernel with rigorous error bounds and a new error con- trol scheme integrating any arbitrary approx- imation method - within the best discrete- algorithmic framework using adaptive hier- archical data structures. We rigorously eval- uate these techniques empirically in the con- text of optimal bandwidth selection in kernel density estimation, revealing the strengths and weaknesses of current state-of-the-art approaches for the rst time. Our re- sults demonstrate that the new error con- trol scheme yields improved performance, whereas the series expansion approach is only eectiv e in low dimensions (v e or less).