Fast Exact Evaluation of Univariate Kernel Sums

This paper presents new methodology for computationally efficient evaluation of univariate kernel sums. It is shown that a rich class of kernels allows for exact evaluation of functions expressed as a sum of kernels using simple recursions. Given an ordered sample the computational complexity is linear in the sample size. Direct applications to the estimation of denisties and their derivatives shows that the proposed approach is competitive with the state-of-the-art. Extensions to multivariate problems including independent component analysis and spatial smoothing illustrate the versatility of univariate kernel estimators, and highlight the efficiency and accuracy of the proposed approach. Multiple applications in image processing, including image deconvolution; denoising; and reconstruction are considered, showing that the proposed approach offers very promising potential in these fields.