An Exponential Lower Bound for Spectral Density Estimation on Unweighted Graphs

2026-06-26

AI Summary

This work investigates the sample complexity lower bound for estimating the spectral density of the normalized adjacency matrix of unweighted graphs via random walks. Addressing a long-standing open problem, it extends— for the first time—the exponential lower bound previously established for weighted graphs to the unweighted setting. By integrating tools from information theory, probabilistic methods, approximation theory under the Wasserstein-1 distance, and careful analysis of random walk trajectories, the study proves that any algorithm achieving an ε-approximation of the spectral density with constant success probability requires at least $2^{\Omega(1/\varepsilon^{1/6})}$ random walk trajectories of comparable length. This result reveals the inherent difficulty of high-precision spectral density estimation on unweighted graphs.

Abstract

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.

Problem

• spectral density estimation
• unweighted graphs
• lower bound
• random walks
• normalized adjacency matrix

Innovation

• spectral density estimation
• exponential lower bound
• unweighted graphs
• random walks
• normalized adjacency matrix