From Dominant Eigenspace Computation to Orthogonal Constrained Optimization Problems
Xin Liu
Prof.
Academy of Mathematics and Systems Science, CAS
Abstract:Recently, identifying dominant eigenvalues or singular values of a sequence of closely related matrices has become an indispensable algorithmic component for many first-order optimization methods for various convex optimization problems, such as semidefinite programming, low-rank matrix completion, robust principal component analysis, sparse inverse covariance matrix estimation, nearest correlation matrix estimation, and so on. More often than not, the computation of the dominant eigenspace forms a major bottleneck in the overall efficiency of solution processes. The well-known Krylov-subspace type of methods have a few limitations including lack of scalability. Since the dominant eigenspace computation can be formulated as a special orthogonal constrained optimization problem, we propose a few optimization based approaches which perform robustly and efficiently in a wide range of scenarios. Moreover, we study the general orthogonal constrained optimization problems and propose a new algorithm framework in which either Stiefel manifold or its tangent space related calculations are waived. Numerical performance illustrates the great potential of the new algorithms based on this framework.
Bio: 2004年本科毕业于北京大学数学科学学院； 2009年于中国科学院研究生院获得博士学位，导师是袁亚湘院士。毕业后留所工作至今。期间分别在德国ZIB研究所、美国 RICE大学、美国纽约大学Courant研究所进行过长期访问。主要研究方向包括正交约束矩阵优化问题，包括线性与非线性特征值问题；非线性最小二乘问的算法与理论；分布式优化算法设计。刘歆主持并完成一项国家自然科学基金青年基金项目；现主持一项国家自然科学基金面上项目，并千2016年8月获得国家自然科学基金委优秀青年科学基金。2014年12 月入选中国科学院数学中国运筹学会青年科技奖； 2017年2月入选中国科学院北京分院“启明星”优秀人才计划。于2015 年7月起担任《 Mathematical Programming Computation 》编委， 千2017年7月起担任《计算数学》编委。