Moments, Random Walks, and Limits for Spectrum Approximation

TL;DR

This paper demonstrates fundamental limits in approximating distributions from moments, showing that even with extensive spectral information, certain distributions cannot be accurately reconstructed in Wasserstein-1 distance.

Contribution

It establishes lower bounds on distribution approximation from moments and spectral data, matching existing upper bounds and highlighting algorithmic limitations.

Findings

Distributions on [-1,1] cannot be approximated within epsilon in Wasserstein-1 distance despite knowing all moments.

Spectral approximation of graph eigenvalues has inherent complexity barriers, requiring exponential resources.

Improving existing spectral approximation algorithms would necessitate fundamentally new approaches.

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.