An Exponential Lower Bound for Spectral Density Estimation on Unweighted Graphs

We study lower bounds for estimating the spectral density of the normalized adjacency matrix of a graph. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ random walks initiated from uniformly random nodes in the graph. Later, Jin et al. [COLT 2023] established a nearly matching exponential lower bound for \emph{weighted} graphs, assuming the algorithm has access to samples from random walks started at random nodes. It was left open whether this lower bound could be extended to \emph{unweighted} graphs. In this paper, we answer this question in the affirmative by proving an exponential lower bound for unweighted graphs. Specifically, we show that no algorithm can compute an $\varepsilon$-approximation to the spectrum of a normalized graph adjacency matrix with constant success probability, even when given the full transcripts of $2^{Ω(1/\varepsilon^{1/6})}$ random walks, each of length $2^{Ω(1/\varepsilon^{1/6})}$, started from uniformly random nodes.