Dynamic Kernel Graph Sparsifiers

Yang Cao, Wenyu Jin, Xiaoyu Li, Zhao Song, Xiaorui Sun, OMRI WEINSTEIN

Keywords: Kernel, Geometric Graph, Theory

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 $P$.

Primary Area: optimization

Submission Number: 14580