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题目: Geometry in Solving Systems of Quadratic Equations




时间:2019年5月17日 下午1:30




摘要: Systems of quadratic equations arise from many fundamental problems in data science and imaging. As an example, we consider the problem of solving systems of phaseless equations |<a_i,x>|^2=y_i, i=1,…,m and x in R^n is unknown. One application of great importance is the phase retrieval problem, which provides promising and indispensable tools in a wide spectrum of techniques including X-ray crystallography, diffraction imaging, microscopy, and even quantum mechanics. We will present two results related to geometry in solving systems of phaseless equations.  
(1) We will present a Riemannian gradient descent algorithm and a truncated variant. The algorithms are developed by exploiting the inherent low rank structure of the problem based on the embedded manifold of rank-1 positive semidefinite matrices. Under a random Gaussian model, theoretical recovery guarantee has been established for the truncated variant, showing that the algorithm is able to achieve successful recovery when the number of equations is proportional to the number of unknowns. 
(2) We will present the global geometry of a new non-convex objective function for solving the phaseless equations. The new objective function is constructed via a least squares fitting to the phaseless equations with an activation function. We prove that this new objective function has a nice geometry --- there is no spurious local minima --- when the number of equations is proportional to the number of unknowns. Again, the analysis is done under a random Gaussian model. Therefore, any algorithm finding a local minimum will give a solution of the phaseless equation. Similar geometric results for other systems of quadratic equations were provided by us or other authors under various settings.


报告人简介:蔡剑锋,香港科技大学数学系副教授。2007年在香港中文大学数学系获得博士学位。博士毕业后先后在新加坡国立大学淡马锡实验室任研究员(2007-2009),美国加州大学洛杉矶分校数学系任CAM助理教授(2009-2011),美国爱荷华大学数学系任助理教授(2011-2015)。2015年入职香港科技大学数学系任副教授。主要研究领域是计算调和分析、优化、数值线性代数、高维概率,及其在数据科学与成像技术中的应用。在理论上,将图像领域独立发展近30年的两个数学分支(PDE变分方法和小波方法)建立深刻联系。在算法开发上,提出的奇异值阈值算法已成为低秩矩阵补全的基本工具。在应用上,为盲解卷积、核磁共振波谱信号重构等实际问题提供了行之有效的解决方案。蔡剑锋在包括《Journal of AMS》,《Applied and Computational Harmonic Analysis》,《SIAM 系列期刊》,《IEEE Transactions 系列期刊》,《Journal of Machine Learning Research》,《CVPR》,《ICCV》等国际重要学术期刊和会议上发表论文60多篇。现任《Statistics, Optimization & Information Computing》领域主编及《Mathematics, Computation and Geometry of Data》编委。是2017年和2018年全球高被引科学家之一。  


邀请人:陶敏 老师