High-Dimensional Probability for High-Dimensional Data

人工智能正在经历一场革命，这场革命是由深度学习的经验成功推动的，深度学习是一类基于人工神经网络的机器学习方法。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的大数据学习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，该项目将启发未来的理论发展，在高维概率，特别是在随机矩阵理论。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，对数学论证的迫切需求是显而易见的，其在现实世界数据应用中的惊人成功尚未得到理论解释。该项目提出将高维概率作为深度学习数学探索的自然框架。在这个建议中的大多数问题的一个统一的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。非线性赋予神经网络和量化器以力量，是随机布尔阈值函数、随机张量和几何图形概念的基础。探索伪线性和非齐次随机矩阵的不寻常谱行为是该项目的主要目标。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。

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.