题目: Rate-optimal Perturbation Bounds for Singular Spaces, SVD and PCA
 

报告人：张安如 （威斯康星大学-麦迪逊分校） 

时间： 2019年12月26号 16:30-17:30 

地点：北航沙河主楼E405

摘要:  Perturbation bounds for singular spaces, principal component analysis (PCA) and singular value decomposition (SVD) are fundamental tools and have been widely used in statistics, machine learning, and applied mathematics. In this talk, we discuss two recent results on perturbation bounds, SVD, PCA, and their applications.

      First, we establish separate perturbation bounds, measured in both spectral and Frobenius sin-Theta distances, for the left and right singular subspaces. Lower bounds, which show that the individual perturbation bounds are rate-optimal, are also given. The new perturbation bounds are applicable to a wide range of problems. In this paper, we consider in detail applications to low-rank matrix denoising and singular space estimation, high-dimensional clustering, and canonical correlation analysis (CCA).

      Second, we consider PCA and SVD in the presence of heteroskedastic noise, which arises naturally in a range of applications. We introduce a general framework for heteroskedastic PCA and propose an algorithm called HeteroPCA, which involves iteratively imputing the diagonal entries to remove the bias due to heteroskedasticity. This procedure is computationally efficient and provably optimal under the generalized spiked covariance model. A key technical step is a deterministic robust perturbation analysis on the singular subspace, which can be of independent interest.

      报告人简介:  张安如，威斯康星大学-麦迪逊分校助理教授。2010年毕业于北京大学数学学院。2015年于宾夕法尼亚大学获得博士学位，师从国际著名统计学家Tony Cai教授。张安如博士的研究领域涉及高维数据的统计推断、统计学习理论以及优化方法等。他已经在Annals of Statistics, JASA，Biometrika和IEEE Transactions on Information Theory等一系列国际权威期刊发表二十余篇文章。

邀请人：陈迪荣