报告人：凌舒扬（上海纽约大学）

时  间：7月16日下午15:00

地  点：厦大海韵园实验楼105报告厅

内容摘要:

Given a set of multiple point clouds, how to find the rigid transformations (rotation, reflection, and shifting) such that these point clouds are well aligned? This problem, known as the generalized orthogonal Procrustes problem (GOPP), plays a fundamental role in several scientific disciplines including statistics, imaging science and computer vision. Despite its tremendous practical importance, it is still a challenging computational problem due to the inherent nonconvexity. In this paper, we study the semidefinite programming (SDP) relaxation of the generalized orthogonal Procrustes problems and prove that the tightness of the SDP relaxation holds, i.e., the SDP estimator exactly equals the least squares estimator, if the signal-to-noise ratio (SNR) is relatively large. We also prove that an efficient generalized power method with a proper initialization enjoys global linear convergence to the least squares estimator. In addition, we analyze the Burer-Monteiro factorization and show the corresponding optimization landscape is free of spurious local optima if the SNR is large. This explains why first-order Riemannian gradient methods with random initializations usually produce a satisfactory solution despite the nonconvexity. One highlight of our work is that the theoretical guarantees are purely algebraic and do not require any assumptions on the statistical property of the noise. Our results partially resolve one open problem posed in [Bandeira, Khoo, Singer, 2015] on the tightness of the SDP relaxation in solving the generalized orthogonal Procrustes problem. Numerical simulations are provided to complement our theoretical analysis.

个人简介：

凌舒扬现任上海纽约大学数据科学助理教授。他2017年获加州大学戴维斯分校应用数学博士学位，2017-2019年在纽约大学柯朗数学研究所和数据科学中心担任柯朗讲师/助理教授。他的研究涉及优化、概率、统计、计算谐波分析和数值线性代数等，已有多篇论文发表在知名学术期刊如Math. Program, Inverse Problems, SIAM J. Optim.等上。

联系人：杜魁