Can I read A Quasi-Monte Carlo Data Structure for Smooth Kernel Evaluations on EtoBox?

A Quasi-Monte Carlo Data Structure for Smooth Kernel Evaluations by Charikar, Moses; Kapralov, Michael; Waingarten, Erik is a scholarly article available to read on EtoBox.

What is A Quasi-Monte Carlo Data Structure for Smooth Kernel Evaluations about?

In the kernel density estimation (KDE) problem one is given a kernel $K(x, y)$ and a dataset $P$ of points in a Euclidean space, and must prepare a data structure that can quickly answer density queries: given a point $q$, output a $(1+\epsilon)$-approximation to $\mu:=\frac1{|P|}\sum_{p\in P} K(p, q)$. The classical approach to KDE is the celebrated fast multipole method of [Greengard and Rokhlin]. The fast multipole method combines a basic space partitioning approach with a multidimensional Taylor expansion, which yields a $\approx \log^d (n/\epsilon)$ query time (exponential in the dimension $d$). A recent line of work initiated by [Charikar and Siminelakis] achieved polynomial dependence on $d$ via a combination of random sampling and randomized space partitioning, with [Backurs et al.] giving an efficient data structure with query time $\approx \mathrm{poly}{\log(1/\mu)}/\epsilon^2$ for smooth kernels. Quadratic dependence on $\epsilon$, inherent to the sampling methods, is prohibitively expensive for small $\epsilon$. This issue is addressed by quasi-Monte Carlo methods in numerical analysis. The high level idea in quasi-Monte Carlo methods is to replace random sampling with

Author

Charikar, Moses; Kapralov, Michael; Waingarten, Erik

Published

2024

Language

EN