We present a dynamic algorithm for maintaining $(1+\epsilon)$-approximate maximum eigenvector and eigenvalue of a positive semi-definite matrix $A$ undergoing \emph{decreasing} updates, i.e., updates which may only decrease eigenvalues. Given a vector $v$ updating $A\gets A-vv^{\top}$, our algorithm takes $\tilde{O}(\mathrm{nnz}(v))$ amortized update time, i.e., polylogarithmic per non-zeros in the update vector. Our technique is based on a novel analysis of the influential power method in the dynamic setting. The two previous sets of techniques have the following drawbacks (1) algebraic techniques can maintain exact solutions but their update time is at least polynomial per non-zeros, and (2) sketching techniques admit polylogarithmic update time but suffer from a crude additive approximation. Our algorithm exploits an oblivious adversary. Interestingly, we show that any algorithm with polylogarithmic update time per non-zeros that works against an adaptive adversary and satisfies an additional natural property would imply a breakthrough for checking psd-ness of matrices in $\tilde{O}(n^{2})$ time, instead of $O(n^{\omega})$ time.

翻译：我们提出一种动态算法，用于维护正半定矩阵 $A$ 在经历单调递减更新（即仅能降低特征值的更新）时的 $(1+\epsilon)$-近似最大特征向量与特征值。给定更新向量 $v$ 执行 $A\gets A-vv^{\top}$ 操作后，该算法的均摊更新时间为 $\tilde{O}(\mathrm{nnz}(v))$，即与更新向量中非零元素个数呈多对数关系。我们的技术基于对动态场景下经典幂迭代法的新颖分析。现有两类技术存在以下缺陷：(1) 代数方法虽能保持精确解，但每个非零元素的更新时间至少为多项式复杂度；(2) 草图方法虽支持多对数级更新时间，但会引入粗糙的加性近似误差。本算法依赖于非适应性对手假设。值得注意的是，我们证明：若存在任意算法能在每个非零元素的多对数级更新时间内对抗适应性对手，且满足另一自然性质，则将直接推动矩阵半正定性检验问题的突破——其时间复杂度将从原有的 $O(n^{\omega})$ 降至 $\tilde{O}(n^{2})$。