基于Kaczmarz方法的相位恢复及其理论研究

Data science is a cutting-edge interdiscipline closely related to machine learning, statistics and optimization. Learning theory provides a mathematical framework to investigate the theoretical properties of machine learning algorithms and plays an important role in the era of big data. Phase retrieval aims at recovering an unknown signal from its magnitude-only measurements, which attracts more and more attentions due to its successful applications in X-ray crystallography, microscopy, optical imaging and so on. In this project we focus on designing and analyzing Kaczmarz type learning algorithms for large-scale and high-dimensional (even infinite dimensional) phase retrieval problems. We shall first consider the relaxed randomized Kaczmarz method for noisy phase retrieval setting, where convergence analysis for both real and complex case will be conducted. We shall then study the impact of sparsity on the randomized sparse Kaczmarz method for phase retrieval and derive its learning rates. This helps to deal with high dimensional phase retrieval problems. At last, we shall investigate the theoretical properties of Kaczmarz methods in general infinite dimensional Hilbert space for both noiseless and noisy phase retrieval problems.

数据科学是一个与机器学习、统计学以及优化都密切相关的前沿交叉学科。学习理论为机器学习算法的理论研究提供了一个良好的数学框架，在如今的大数据时代起着至关重要的作用。相位恢复旨在仅依靠信号的强度信息恢复其相位信息，由于它在x射线晶体学、显微镜学、光学成像学等领域的成功应用而吸引了越来越多的关注。此项目中我们主要考虑的是如何在大规模、高维(甚至无穷维)相位恢复问题中设计和分析Kaczmarz式学习算法。我们将首先考虑有噪声情形相位恢复问题中的松弛随机Kaczmarz方法并对实数域和复数域两种情况分别进行收敛性分析。然后我们将研究相位恢复问题中稀疏性对随机稀疏Kaczmarz方法的影响并得到其收敛速度，这有利于解决高维相位恢复问题。最后，我们将探讨Kaczmarz方法在一般无穷维希尔伯特空间相位恢复的理论性质，包括无噪声和有噪声两种情况。

学习理论提供各种工具以建立现代机器学习的数学基础并设计有效算法来处理和分析现实世界的复杂数据。本项目主要在再生核希尔伯特空间的框架内研究大规模和高维（甚至无穷维）问题。在理论方面，我们建立学习算法泛化误差的最优收敛速度。在计算方面，我们研究了它们在分布式学习，在线学习或者鲁棒学习设定下的一致性和有效率。在本项目资助下，我们已在应用数学和机器学习领域的国际知名期刊，如Journal of Machine Learning Research, Analysis and Applications, Communications on Pure and Applied Analysis等杂志上发表多篇高质量论文。