The Method of Moments [Pea94] is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or non-uniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analysis of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions. This paper considers estimating a $k$-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensures existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for Generalized Method of Moments [Han82] which involves non-convex optimization and lacks theoretical guarantees.

中文翻译：

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通过矩的去噪方法对高斯混合的最优估计

矩量法 [Pea94] 是统计学中使用最广泛的参数估计方法之一，它通过求解匹配总体和估计矩的方程组。然而，在实践中，特别是对于混合模型的重要情况，人们经常需要应对具有统计意义的解不存在或不唯一的困难，以及求解大型多项式系统的高计算成本。此外，矩量法的理论分析主要局限于强假设下建立的结果的渐近正态形式。本文考虑用一个共同的（可能未知的）方差参数来估计一个 $k$ 分量的高斯位置混合。为了克服上述理论和算法障碍，关键的一步是在求解矩方程方法之前，通过投影到截断矩空间（通过半定规划）来对矩估计进行去噪。这种正则化不仅确保了解的存在性和唯一性，而且还通过高斯求积产生了快速求解器。此外，通过多项式插值和专业化技术在 Wasserstein 距离中证明新的矩比较定理，我们建立了所提出程序的统计保证和自适应最优性，以及错误指定模型中的预言不等式。这些结果也可以被视为广义矩法 [Han82] 的可证明算法，该算法涉及非凸优化并且缺乏理论保证。