CRII: III: Efficient and Robust Statistical Estimation from Nonlinear Compressed Measurements

该项目通过为从高维数据中发现知识提供新的理论和算法，推动了国家在科学和工程方面的发展。高维估计是一种从大量冗余或不相关特征中提取最有用信息的计算过程，在医学成像、生物学和气候学等各个领域都发挥了重要作用。然而，当数据具有复杂的结构时，或者当它们由于硬件故障、编程错误或网络攻击而被污染时，成熟的估计方案会急剧下降。该项目的目标是显着扩大学习算法对不同类型的结构和数据错误的基本限制的理解，为鲁棒算法设计提供完整的指导方针，并突出智能系统的可靠性和一致性。本项目将通过研究高维统计、优化和学习理论之间的相互作用，为机器学习、高维统计、信号处理、生物学等相关领域的研究人员提供理论成果、算法实现和可重用的经验数据等支持。研究人员将开发一个统一的框架，用于高维区域中的非线性估计，该框架揭示了量化测量的参数估计和深度神经网络中的非线性激活函数的学习。特别是，考虑到非线性和可能的非凸性质，研究人员将开发有效的约束优化算法，利用固有的几何结构到算法设计和理论分析。基于统一的框架和建立的通用结果，调查员将重新审视启发式算法的合奏，并将提供一个理论上的理由，何时以及为什么他们在实践中取得成功。最后，研究人员将设计对各种类型的数据损坏（如对抗性噪声，离群值和恶意噪声）具有鲁棒性的算法。为了在样本复杂性中获得对噪声率和数据维度的接近最佳的依赖性，将利用学习理论和稳健统计学中的工具和丰富的理论来建立一系列新的统计结果。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project advances the nation's development in science and engineering by providing new theory and algorithms for knowledge discovery from high-dimensional data. High-dimensional estimation, a computational procedure that extracts the most useful information from a large pool of redundant or irrelevant features, has played fundamental roles in various areas such as medical imaging, biology, and climatology. However, the well-established estimation schemes degrade dramatically when the data have complex structures, or when they are contaminated due to hardware failures, programming errors, or cyber-attacks. The goal of this project is to significantly broaden the understanding of the fundamental limits of learning algorithms against different types of structures and data errors, to offer a complete guideline for robust algorithmic design, and to highlight the extent to which an intelligent system behaves reliably and consistently. Outputs, such as theoretical results, algorithm implementation, and reusable empirical data, are designed to support a wide range of researchers in machine learning, high-dimensional statistics, signal processing, biology, and other related fields.The project will be carried out by investigating the interplay of high-dimensional statistics, optimization, and learning theory. The investigator will develop a unified framework for nonlinear estimation in the high-dimensional regime, which uncovers parameter estimation from quantized measurements and learning with nonlinear activation functions in deep neural networks. In particular, to account for the nonlinear and possibly nonconvex nature, the investigator will develop efficient constrained optimization algorithms by leveraging inherent geometric structures into algorithmic design and theoretical analysis. Based on the unified framework and the established generic results, the investigator will revisit an ensemble of heuristic algorithms and will provide a theoretical justification on when and why they succeed in practice. Lastly, the investigator will design algorithms that are robust to various types of data corruption, such as adversarial noise, outlier, and malicious noise. To obtain a near-optimal dependence on the noise rate and data dimension in the sample complexity, a series of new statistical results will be established by leveraging tools from, and enriching theory in learning theory and robust statistics.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.