Constrained Statistical Estimation and Inference: Theory, Algorithms and Applications

这个项目处于统计学和机器学习的边缘。其基本主题是利用在复杂的科学数据分析问题中存在的限制，但这些限制在传统方法中尚未得到彻底研究。该项目将探索统计程序的理论、算法和应用，并对估计器和应用程序的存储、运行时间、形状、能量或物理施加限制。这项研究的总体目标是开发能够帮助科学家进行更有效的数据分析的理论和工具。许多统计方法纯粹是“数据驱动”的，只对底层模型施加平稳性或规律性限制。特别是，经典统计理论研究估计量，而不考虑它们的计算要求。在现代数据分析环境中，包括天文学、云计算和嵌入式设备，计算需求通常是核心。该项目将开发极小极大理论和算法，用于在存储、计算和能量限制下的非参数估计和检测问题。其他需要研究的约束包括高维数据的凸性和单调性等形状限制。该项目还将调查通过使用偏微分方程以及物理动力学和力学模型纳入物理约束的情况，重点是算法和理论界限。

This project lies at the boundary of statistics and machine learning. The underlying theme is to exploit constraints that are present in complex scientific data analysis problems, but that have not been thoroughly studied in traditional approaches. The project will explore theory, algorithms, and applications of statistical procedures, with constraints imposed on the storage, runtime, shape, energy or physics of the estimators and applications. The overall goal of the research is to develop theory and tools that can help scientists to conduct more effective data analysis.Many statistical methods are purely "data driven" and only place smoothness or regularity restrictions on the underlying model. In particular, classical statistical theory studies estimators without regard to their computational requirements. In modern data analysis settings, including astronomy, cloud computing, and embedded devices, computational demands are often central. The project will develop minimax theory and algorithms for nonparametric estimation and detection problems under constraints on storage, computation, and energy. Other constraints to be studied include shape restrictions such as convexity and monotonicity for high dimensional data. The project will also investigate the incorporation of physical constraints through the use of PDEs and models of physical dynamics and mechanics, focusing on both algorithms and theoretical bounds.