Dynamic Kernel Graph Sparsifiers

TL;DR

This paper introduces a dynamic data structure for maintaining spectral sparsifiers of geometric graphs efficiently as points move, with applications in optimization and matrix computations.

Contribution

It presents the first fully-dynamic spectral sparsifier for geometric graphs with sublinear update time and robustness against adaptive adversaries.

Findings

Update time is $n^{o(1)}$ with high probability.

Initialization time is $n^{1+o(1)}$.

Supports efficient matrix-vector operations for geometric graph Laplacians.

Abstract

A geometric graph associated with a set of points $P= \{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$ 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 $n^{o(1)}$ with high probability, and the initialization time is $n^{1+o(1)}$. Under certain assumption, our data structure can be made robust against adaptive adversaries, which makes our sparsifier applicable in iterative optimization algorithms. We further show that the Laplacian matrices corresponding to geometric graphs admit a randomized sketch for maintaining matrix-vector multiplication and projection in $n^{o(1)}$ time, under sparse updates to the query vectors, or under modification of points in $P$.