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

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.

利用半定规划抑噪，解决高斯混合模型中方法矩的不一致性等问题，建立在 Wasserstein 距离下新的矩比较定理，提出新的算法，解决方法矩估计常见的问题。