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High-Dimensional Probability for High-Dimensional Data

高维数据的高维概率

基本信息

批准号：
1954233
负责人：
Roman Vershynin
金额：
$ 36万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2024-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954233&HistoricalAwards=false
关键词：
Dimensional Probability Data

项目摘要

Artificial Intelligence is undergoing a revolution that is fueled by empirical successes of deep learning, a class of machine learning methods based on artificial neural networks. These successes reverberate across a broad spectrum of data science problems. Nevertheless, theoretical understanding of deep learning is scarce. This project is aimed at building rigorous mathematical foundations for modern and future approaches to learning from big data. High-dimensional probability is proposed as a natural framework for the mathematical exploration of deep learning. This project has a double benefit. On the one hand, it is aimed at theoretically explaining the successes of deep learning. On the other hand, the project will inspire future theoretical developments in high-dimensional probability, especially in random matrix theory. The project also provides research training opportunities for graduate students. This project will address theoretical problems in high-dimensional probability that are inspired by open problems in data science. A pressing need for mathematical justification is evident in the area of deep learning, whose stunning success on real-world data applications is not theoretically explained yet. This project proposes high-dimensional probability as a natural framework for the mathematical exploration of deep learning. A unifying theme of most of the problems in this proposal is nonlinear random matrix theory, where random matrices are transformed by a nonlinearity, which alters their spectral and geometric behavior. Nonlinearities empower neural networks and quantizers, underlie the concepts of random Boolean threshold functions, random tensors and geometric graphs. Exploring the unusual spectral behavior of pseudolinear and inhomogeoenous random matrices are the main general thrust of this project.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.

深度学习是一种基于人工神经网络的机器学习方法，在深度学习的经验成功推动下，人工智能正在经历一场革命。这些成功在广泛的数据科学问题中产生了反响。然而，对深度学习的理论理解是稀缺的。该项目旨在为现代和未来从大数据中学习的方法建立严格的数学基础。高维概率被提出作为深度学习数学探索的自然框架。这个项目有双重好处。一方面，它旨在从理论上解释深度学习的成功。另一方面，该项目将对高维概率，特别是随机矩阵理论的未来理论发展产生启发。该项目还为研究生提供研究培训机会。该项目将解决高维概率中的理论问题，这些问题受到数据科学中开放问题的启发。在深度学习领域，对数学证明的迫切需求是显而易见的，深度学习在现实世界数据应用上的惊人成功还没有从理论上得到解释。该项目提出高维概率作为深度学习数学探索的自然框架。这个建议中大多数问题的一个统一主题是非线性随机矩阵理论，其中随机矩阵被非线性变换，这改变了它们的光谱和几何行为。非线性增强了神经网络和量化器的能力，是随机布尔阈值函数、随机张量和几何图的基础。探索伪线性和非齐次随机矩阵的异常光谱行为是本项目的主要总推力。该奖项反映了美国国家科学基金会的法定使命，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。