黎曼流形上的无约束优化及其应用

Unconstrained optimization on Riemannian manifold with applications

流形上的优化算法广泛应用于重要的科学工程问题,包括但不限于图像处理、信号复原、机器学习、网络分析、计算机视觉与图形学等,近些年受到广泛关注,并吸引许多知名学者在该领域开展研究工作.本文首先介绍流形优化算法设计中所需要的几何工具,再从光滑与非光滑两个角度回顾流形上无约束优化领域的重要进展,然后通过介绍流形优化算法在重要的科学工程领域的应用说明流形优化的应用背景,最后结合现有的流形优化进展展望未来的流形优化研究方向.

Optimization algorithms on manifolds are widely used in important scientific and engineering problems,including but not limited to image processing,signal restoration,machine learning,network analysis,computer vision and graphics among others.In recent years,these algorithms have intensely attracted the research attention,and numerous prominent scholars have conducted research work in this field.In this article,we first summarize geometric tools required in the design of manifold optimization algorithms and then review the important progress in the field of smooth unconstrained optimization on manifolds from both smooth and nonsmooth perspectives.Then,via introducing the application of manifold optimization algorithms in important scientific and engineering fields,the application background of manifold optimization is explained.Finally,based on the existing progress in manifold optimization,future research directions in manifold optimization are prospected.