We consider the problem of estimating the spectral density of the normalized adjacency matrix of an $n$-node undirected graph. We provide a randomized algorithm that, with $O(n\epsilon^{-2})$ queries to a degree and neighbor oracle and in $O(n\epsilon^{-3})$ time, estimates the spectrum up to $\epsilon$ accuracy in the Wasserstein-1 metric. This improves on previous state-of-the-art methods, including an $O(n\epsilon^{-7})$ time algorithm from [Braverman et al., STOC 2022] and, for sufficiently small $\epsilon$, a $2^{O(\epsilon^{-1})}$ time method from [Cohen-Steiner et al., KDD 2018]. To achieve this result, we introduce a new notion of graph sparsification, which we call nuclear sparsification. We provide an $O(n\epsilon^{-2})$-query and $O(n\epsilon^{-2})$-time algorithm for computing $O(n\epsilon^{-2})$-sparse nuclear sparsifiers. We show that this bound is optimal in both its sparsity and query complexity, and we separate our results from the related notion of additive spectral sparsification. Of independent interest, we show that our sparsification method also yields the first deterministic algorithm for spectral density estimation that scales linearly with $n$ (sublinear in the representation size of the graph).

翻译：我们考虑估计 $n$ 节点无向图归一化邻接矩阵谱密度的问题。我们提出一种随机算法，通过对度和邻居预言机的 $O(n\epsilon^{-2})$ 次查询，并在 $O(n\epsilon^{-3})$ 时间内，以 Wasserstein-1 度量达到 $\epsilon$ 精度的谱估计。这改进了先前的最先进方法，包括 [Braverman 等人，STOC 2022] 的 $O(n\epsilon^{-7})$ 时间算法，以及对于足够小的 $\epsilon$，[Cohen-Steiner 等人，KDD 2018] 的 $2^{O(\epsilon^{-1})}$ 时间方法。为实现这一结果，我们引入一种新的图稀疏化概念，称为核稀疏化。我们提供一种 $O(n\epsilon^{-2})$ 查询和 $O(n\epsilon^{-2})$ 时间的算法，用于计算 $O(n\epsilon^{-2})$ 稀疏的核稀疏化器。我们证明该界限在稀疏性和查询复杂度上均是最优的，并将我们的结果与相关的加性谱稀疏化概念区分开来。值得独立关注的是，我们表明我们的稀疏化方法还产生了首个与 $n$ 线性缩放（即在图表示大小下呈次线性）的谱密度估计确定性算法。