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。