论文标题
关于混合模型中非参数最大似然估计量的有效且可扩展的计算
On Efficient and Scalable Computation of the Nonparametric Maximum Likelihood Estimator in Mixture Models
论文作者
论文摘要
在本文中，我们研究了多元混合物模型中非参数最大似然估计器（NPMLE）的计算。我们的第一种方法通过固定NPMLE的支持点并在混合物比例上进行优化，可以离散这个无限的尺寸凸优化问题。在这种情况下，我们提出，利用解决方案的稀疏性，这是一种有效且可扩展的半齿牛顿的增强拉格朗日方法（ALM）。我们的算法击败了最先进的方法〜\ cite {koenker2017rebayes，kim2020fast}，并且可以使用$ n \ 10^6 $数据点，带有$ m \ 10^4 $支持点。我们的第二个程序将预期最大化（EM）算法与上面的ALM方法结合在一起，允许对支持点和概率权重进行联合优化。对于我们的两种算法，我们都为其（超线性）收敛属性提供了形式的结果。 计算出的NPMLE可立即用于剥夺经验贝叶斯框架中的观察结果。在这种情况下，我们提出了新的DeNoise估计，以及它们一致的估计。进行了广泛的数值实验，以说明我们方法的有效性。 In particular, we employ our procedures to analyze two astronomy data sets: (i) Gaia-TGAS Catalog~\cite{anderson2018improving} containing $n \approx 1.4 \times 10^6$ data points in two dimensions, and (ii) the $d=19$ dimensional data set from the APOGEE survey~\cite{majewski2017apache} with $n \大约2.7 \ times 10^4 $。
In this paper we study the computation of the nonparametric maximum likelihood estimator (NPMLE) in multivariate mixture models. Our first approach discretizes this infinite dimensional convex optimization problem by fixing the support points of the NPMLE and optimizing over the mixture proportions. In this context we propose, leveraging the sparsity of the solution, an efficient and scalable semismooth Newton based augmented Lagrangian method (ALM). Our algorithm beats the state-of-the-art methods~\cite{koenker2017rebayes, kim2020fast} and can handle $n \approx 10^6$ data points with $m \approx 10^4$ support points. Our second procedure, which combines the expectation-maximization (EM) algorithm with the ALM approach above, allows for joint optimization of both the support points and the probability weights. For both our algorithms we provide formal results on their (superlinear) convergence properties. The computed NPMLE can be immediately used for denoising the observations in the framework of empirical Bayes. We propose new denoising estimands in this context along with their consistent estimates. Extensive numerical experiments are conducted to illustrate the effectiveness of our methods. In particular, we employ our procedures to analyze two astronomy data sets: (i) Gaia-TGAS Catalog~\cite{anderson2018improving} containing $n \approx 1.4 \times 10^6$ data points in two dimensions, and (ii) the $d=19$ dimensional data set from the APOGEE survey~\cite{majewski2017apache} with $n \approx 2.7 \times 10^4$.
