We consider the problem of approximating a general Gaussian location mixture
by finite mixtures. The minimum order of finite mixtures that achieve a
prescribed accuracy (measured by various $f$-divergences) is determined within
constant factors for the family of mixing distributions with compactly support
or appropriate assumptions on the tail probability including subgaussian and
subexponential. While the upper bound is achieved using the technique of local
moment matching, the lower bound is established by relating the best
approximation error to the low-rank approximation of certain trigonometric
moment matrices, followed by a refined spectral analysis of their minimum
eigenvalue. In the case of Gaussian mixing distributions, this result corrects
a previous lower bound in [Allerton Conference 48 (2010) 620-628].

通过有限混合物逼近一般的高斯位置混合物，在具有紧支持或适当的尾部概率假设的混合分布族中确定实现所需精度（通过各种 $f$- 分歧度测量）的有限混合物的最小阶数，其中上界是通过使用局部矩匹配技术实现的，下界是通过将最佳逼近误差与某些三角矩矩阵的低秩逼近联系起来，然后通过对其最小特征值进行精细的谱分析确定的。在高斯混合分布情况下，该结果纠正了之前在 [Allerton Conference 48 (2010) 620-628] 中的下界。

有限高斯混合逼近的最佳近似

On the best approximation by finite Gaussian mixtures

This paper studies identifiability and convergence behaviors for parameters
of multiple types in finite mixtures, and the effects of model fitting with
extra mixing components. First, we present a general theory for strong
identifiability, which extends from the previous work of Nguyen [2013] and Chen
[1995] to address a broad range of mixture models and to handle matrix-variate
parameters. These models are shown to share the same Wasserstein distance based
optimal rates of convergence for the space of mixing distributions ---
$n^{-1/2}$ under $W_1$ for the exact-fitted and $n^{-1/4}$ under $W_2$ for the
over-fitted setting, where $n$ is the sample size. This theory, however, is not
applicable to several important model classes, including location-scale
multivariate Gaussian mixtures, shape-scale Gamma mixtures and
location-scale-shape skew-normal mixtures. The second part of this work is
devoted to demonstrating that for these "weakly identifiable" classes,
algebraic structures of the density family play a fundamental role in
determining convergence rates of the model parameters, which display a very
rich spectrum of behaviors. For instance, the optimal rate of parameter
estimation in an over-fitted location-covariance Gaussian mixture is precisely
determined by the order of a solvable system of polynomial equations --- these
rates deteriorate rapidly as more extra components are added to the model. The
established rates for a variety of settings are illustrated by a simulation
study.

本文主要探讨了有限混合物中参数的可识别性和收敛行为，以及使用额外混合组件进行模型拟合的影响，并给出了强可识别性的一般理论，以及对于某些弱可识别性的模型，它们密度函数的代数结构对模型参数的估计收敛率具有决定作用，本文通过模拟研究了多个设置的的收敛率。