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.

翻译：我们研究在给定（含噪声）矩测量值条件下，一维分布近似问题的下界。我们证明，存在定义在$[-1,1]$上的分布，即使已知其所有矩的乘性精度达到$(1\pm2^{-\Omega(1/\epsilon)})$，也无法在Wasserstein-1距离下将其近似至精度$\epsilon$；这一结果与Kong和Valiant [Annals of Statistics, 2017]的上界相匹配。为获得该结果，我们构造了一个硬实例，其中分布由精心构造的图邻接矩阵的特征值谱诱导产生。在Wasserstein-1距离下高效近似此类谱是一个已被充分研究的算法问题，Cohen-Steiner等人[KDD 2018]的最新成果提出了一种方法，该方法基于利用图中均匀随机节点起始的$2^{O(1/\epsilon)}$次随机游走精确近似谱矩。作为我们主要结果的加强，我们证明改善该结果中对$1/\epsilon$的依赖需要全新的算法思路。具体而言，即使给定从随机节点起始的$2^{\Omega(1/\epsilon)}$次长度为$2^{\Omega(1/\epsilon)}$的随机游走记录，也没有算法能够以恒定概率计算出归一化图邻接矩阵谱的$\epsilon$精度近似。