Properties of Approximate Inference for Complex High-Dimensional Models

现代科学数据采集通常会在大样本量的个体上创建具有许多测量值的数据集。 数据中感兴趣的结构可以是低维的或稀疏的。从计量经济学到基因组学、图像和信号处理等科学领域都有这样的例子。 该项目探讨了这种结构的统计推断的近似方法，在一组有代表性的当代设置：高维估计，广义线性混合模型和低秩多变量模型。它旨在开发由最优理论和/或性能保证支持的近似方法。 该项目将为磁共振成像和定量遗传学等特定应用领域的当前问题提供新的理论见解，将经典决策理论的思想引入压缩感知和鲁棒线性建模，严格解决非凸优化问题，并获得比传统凸优化方法更好的重建性能。 它将利用决策理论的思想在提议者的以前的工作提供新的理论见解到一个紧迫的实际问题，在压缩传感-导出适用于磁共振成像和NMR光谱的最佳变密度采样时间表。 该项目还将从理论上研究机器学习中流行的确定性近似推理的一些方法的统计性能，但很少关注一致性，渐近正态性和效率等属性。 该研究将开始具体的例子，在广义线性混合模型的领域，从频率的角度来看，并寻求建立第一类结果的渐近效率的期望传播。 最后，本计画将研究具有低维结构的高度多元模型的特征结构的近似推论。 它将适应詹姆斯的多变量分析的经典框架，以广泛的一类多峰模型。 通过与数量遗传学家的合作，它将开发用于推断高维遗传协方差矩阵中的低维结构的方法。 在这两种情况下，随机矩阵理论的方法将是必不可少的。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Modern scientific data acquisition often creates datasets with many measurements on a large sample size of individuals. The structure of interest in the data may be low dimensional or sparse. Examples arise in scientific domains from econometrics to genomics and image and signal processing and beyond. The project explores approximate methods of statistical inference for such structure in a representative set of contemporary settings: high dimensional estimation, generalized linear mixed models and low rank multivariate models. It aims to develop approximate methods backed by an optimality theory and/or performance guarantees. The work is expected to provide new theoretical insights into current problems in specific application domains such as Magnetic Resonance Imaging and quantitative genetics.The project will bring ideas from classical decision theory to compressed sensing and robust linear modeling, rigorously solving nonconvex optimization problems and obtaining reconstruction performance rigorously better than traditional convex optimization methods. It will exploit decision theoretic ideas in the proposer's previous work to provide new theoretical insights into a pressing practical problem in compressed sensing -- deriving optimal variable-density sampling schedules applicable to Magnetic Resonance Imaging and NMR spectroscopy. The project will also study theoretically the statistical performance of some methods for deterministic approximate inference popular in machine learning, but for which little attention has been given to properties such as consistency, asymptotic normality and efficiency. The study will begin with concrete examples in the realm of generalized linear mixed models, from a frequentist perspective, and seek to establish first-of-kind results for asymptotic efficiency of Expectation Propagation. Finally, the project will study approximate inference for the eigenstructure of highly multivariate models with low dimensional structure. It will adapt James' classical framework for multivariate analysis to a broad class of multispike models. Through collaboration with quantitative geneticists, it will develop methods for inference for low dimensional structure in high dimensional genetic covariance matrices. In both cases, methods from random matrix theory will be essential.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.