CAREER: Semidefinite Programming with Applications in Statistical Learning

Career：半定规划及其在统计学习中的应用这个教职早期职业发展(CALEAR)项目的研究目标是设计一套可伸缩的信息提取算法，可以将大规模数据集转换为稀疏的、因此可解释的模型。许多非常活跃的研究课题，如编码理论中的稀疏恢复，信号处理中的压缩感知和基追踪，统计学中的套索和协方差选择，机器学习中的特征选择，都围绕着这样一个核心思想：寻求稀疏模型是同时稳定统计推理过程和突出底层数据集中结构的一种有意义的方法。更具体地说，这个项目源于统计学习中的两个基本问题。一个是关于变量的选择：一个特定的变量是我们观察建模的关键吗？另一个问题是模型结构：任何两个变量之间的关系是解释这些观察结果的关键吗？本着这种精神，该项目将统计学习和信息论的结果与最新的数学编程技术相结合，以产生稀疏统计估计和解码算法的实际性能界限。从理论的角度来看，这些结果应该有助于阐明统计学中模型一致性和计算复杂性之间的基本权衡。早期的结果清楚地说明了这种权衡在一些特定问题实例中的重要性，但系统的结果很少。统计问题还带来了一系列全新的算法挑战，因为它们需要解决具有相对较粗精度目标的超大规模问题，这与数学规划中的经典假设完全相反。因此，该项目的结果将提高我们在这一背景下对优化算法的理解。从实用的角度来看，高效的稀疏推理算法将使经典统计技术的输出直接被非专家解释，并应帮助我们突出复杂数据集中的关键结构模式。

CAREER: Semidefinite Programming with Applications in Statistical LearningThe research objective of this Faculty Early Career Development (CAREER) project is the design of a set of scalable information extraction algorithms that can turn large-scale data sets into sparse, hence interpretable, models. Many intensely active research topics such as sparse recovery in coding theory, compressed sensing and basis pursuit in signal processing, lasso and covariance selection in statistics, feature selection in machine learning, all revolve around the core idea that seeking sparse models is a meaningful way of simultaneously stabilizing statistical inference procedures, and highlighting structure in the underlying data set. More specifically, this project stems from two fundamental questions in statistical learning. One is about variable selection: Is a particular variable key to the modeling of our observations? The other question is about model structure: Is the relationship between any two variables key to explain these observations? In this spirit, this project combines results in statistical learning and information theory with recent mathematical programming techniques to produce realistic performance bounds on sparse statistical estimation and decoding algorithms. From a theoretical perspective, these results should help shed light on a fundamental tradeoff in statistics between model consistency on one hand and computational complexity on the other. Early results have clearly illustrated the significance of this tradeoff on a few particular problem instances, but systematic results are scarce. Statistical problems also pose an entirely new set of algorithmic challenges as they require solving very large-scale problems with relatively coarse precision targets, which is the exact opposite of classical assumptions in mathematical programming. The results of this project will thus improve our understanding of optimization algorithms in this context. From a practical perspective, efficient sparse inference algorithms will make the output of classic statistical techniques directly interpretable by non-experts and should help us highlight key structural patterns in complex data sets.