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 $P$.

翻译：对于点集 $P= \{x_1, x_2, \cdots, x_n \} \subset \mathbb{R}^d$ 与固定核函数 $\mathsf{K}:\mathbb{R}^d\times \mathbb{R}^d\to\mathbb{R}_{\geq 0}$，其对应的几何图是在 $P$ 上定义的完全图，其中边 $(x_i, x_j)$ 的权重为 $\mathsf{K}(x_i, x_j)$。本文提出一种全动态数据结构，能够在逐点更新 $P$ 中点位置的情况下，维护该几何图的一个谱稀疏化器。该数据结构的更新时间复杂度以高概率为 $n^{o(1)}$，初始化时间复杂度为 $n^{1+o(1)}$。在一定假设下，我们的数据结构能够抵御自适应敌手攻击，从而使得该稀疏化器可应用于迭代优化算法中。我们进一步证明，几何图对应的拉普拉斯矩阵支持一种随机化草图技术，能够在查询向量进行稀疏更新或 $P$ 中点发生修改时，以 $n^{o(1)}$ 的时间复杂度维护矩阵-向量乘法与投影运算。