Faster Spectral Density Estimation and Sparsification in the Nuclear Norm

TL;DR

This paper introduces a faster randomized algorithm for spectral density estimation of graphs, utilizing a novel nuclear sparsification technique that improves accuracy and efficiency over previous methods.

Contribution

The paper presents a new nuclear sparsification method and a faster spectral density estimation algorithm with optimal query complexity and sparsity bounds.

Findings

Achieves spectral density estimation within $ ext{O}(n ext{epsilon}^{-2})$ queries and time.

Introduces nuclear sparsification, a new graph sparsification concept.

Provides the first deterministic linear-scale spectral density estimation algorithm.

Abstract

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).