Properties of Approximate Inference for Complex High-Dimensional Models

现代科学数据采集通常会在大样本个体上创建具有多个测量值的数据集。数据中感兴趣的结构可以是低维的或稀疏的。例子出现在从计量经济学到基因组学、图像和信号处理等科学领域。该项目探索了在一组具有代表性的当代背景下对这种结构进行统计推断的近似方法：高维估计、广义线性混合模型和低阶多变量模型。它的目标是开发有最优性理论和/或性能保证支持的近似方法。这项工作有望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。该项目将把经典决策理论的思想引入压缩感知和稳健线性建模，严格求解非凸优化问题，并获得比传统凸优化方法严格更好的重建性能。它将利用提出者先前工作中的决策理论思想，为压缩传感中的一个紧迫的实际问题提供新的理论见解--推导出适用于磁共振成像和核磁共振光谱学的最优变密度采样方案。该项目还将从理论上研究机器学习中流行的一些确定性近似推理方法的统计性能，但对这些方法的一致性、渐近正态和效率等性质关注较少。这项研究将从广义线性混合模型领域的具体例子开始，从频率学家的角度出发，试图建立期望传播的渐近有效性的第一个结果。最后，该项目将研究高度多变量低维结构模型的特征结构的近似推断。它将使詹姆斯的经典多变量分析框架适用于一大类多变量模型。通过与数量遗传学家的合作，它将开发高维遗传协方差矩阵中低维结构的推断方法。在这两种情况下，来自随机矩阵理论的方法都将是必不可少的。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.