Dynamic Kernel Sparsifiers

Abstract: A geometric graph associated with a set of points $$P= {x_1, x_2, \cdots, x_n } \subset \mathbb{R}<sup>d$$ and a fixed kernel function $$\mathsf{K}:\mathbb{R}^d\times \mathbb{R}^{d\to\mathbb{R}_{\geq} 0}$$ is a complete graph on $P$ such that the weight of edge $(x_i, x_j)$ is $\mathsf{K}(x_i, x_j)$. We present a fully-dynamic data structure that maintains a spectral sparsifier of a geometric graph under updates that change the locations of points in $P$ one at a time. The update time of our data structure is $n<sup>{o(1)}$ with high probability, and the initialization time is $n<sup>{1+o(1)}$. Under certain assumption, we can provide a fully dynamic spectral sparsifier with the robostness to adaptive adversary. We further show that, for the Laplacian matrices of these geometric graphs, it is possible to maintain random sketches for the results of matrix vector multiplication and inverse-matrix vector multiplication in $n<sup>{o(1)}$ time, under updates that change the locations of points in $P$ or change the query vector by a sparse difference.