Moments, Random Walks, and Limits for Spectrum Approximation

• URL: http://arxiv.org/abs/2307.00474v1
• Date: Sun, 2 Jul 2023 05:03:38 GMT
• Title: Moments, Random Walks, and Limits for Spectrum Approximation
• Authors: Yujia Jin and Christopher Musco and Aaron Sidford and Apoorv Vikram Singh
• Abstract summary: We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $epsilon$ in Wasserstein-1 distance. No algorithm can compute an $epsilon$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability.
• Score: 40.43008834125277
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study lower bounds for the problem of approximating a one dimensional distribution given (noisy) measurements of its moments. We show that there are distributions on $[-1,1]$ that cannot be approximated to accuracy $\epsilon$ in Wasserstein-1 distance even if we know \emph{all} of their moments to multiplicative accuracy $(1\pm2^{-\Omega(1/\epsilon)})$; this result matches an upper bound of Kong and Valiant [Annals of Statistics, 2017]. To obtain our result, we provide a hard instance involving distributions induced by the eigenvalue spectra of carefully constructed graph adjacency matrices. Efficiently approximating such spectra in Wasserstein-1 distance is a well-studied algorithmic problem, and a recent result of Cohen-Steiner et al. [KDD 2018] gives a method based on accurately approximating spectral moments using $2^{O(1/\epsilon)}$ random walks initiated at uniformly random nodes in the graph. As a strengthening of our main result, we show that improving the dependence on $1/\epsilon$ in this result would require a new algorithmic approach. Specifically, no algorithm can compute an $\epsilon$-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of $2^{\Omega(1/\epsilon)}$ random walks of length $2^{\Omega(1/\epsilon)}$ started at random nodes.

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