2023/07/02 by Yujia Jin, Jin, Yujia, Christopher Musco +5 · 1 citation
Engineering · Mathematics · #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Machine Learning (cs.LG) #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.2307.00474
openalex publication_date 2023/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ε in Wasserstein-1 distance even if we know all of their moments to multiplicative accuracy (1±2-Ω(1/ε)); 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 2O(1/ε) 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/ε in this result would require a new algorithmic approach. Specifically, no algorithm can compute an ε-accurate approximation to the spectrum of a normalized graph adjacency matrix with constant probability, even when given the transcript of 2Ω(1/ε) random walks of length 2Ω(1/ε) started at random nodes.