报告题目：Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances

主 讲 人：刘歆 研究员（中国科学院数学与系统科学研究院）

报告邀请人：徐丽君 副教授

报 告 时间：2025年11月26日上午8:30-10:30

报告地点：数理楼 224

报告摘要：We consider the distributionally robust optimization (DRO) model of principal component analysis (PCA) to account for uncertainty in the underlying probability distribution. The resulting formulation leads to a nonsmooth constrained min-max optimization problem, where the ambiguity set captures the distributional uncertainty by the type-2 Wasserstein distance. We prove that the inner maximization problem admits a closed-form optimal value. This explicit characterization equivalently reformulates the original DRO model into a minimization problem on the Stiefel manifold with intricate nonsmooth terms, a challenging formulation beyond the reach of existing algorithms. To address this issue, we devise an efficient smoothing manifold proximal gradient algorithm. Our analysis establishes Riemannian gradient consistency and global convergence of our algorithm to a stationary point of the nonsmooth minimization problem. We also provide the iteration complexity O(ϵ−3) of our algorithm to achieve an ϵ-approximate stationary point. Finally, numerical experiments are conducted to validate the effectiveness and scalability of our algorithm, as well as to highlight the necessity and rationality of adopting the DRO model for PCA.

个人简历：刘歆，中国科学院数学与系统科学研究院研究员，博士生导师，计算数学与科学工程计算研究所副所长。主要研究方向包括流形优化、分布式优化及其在材料计算、大数据分析和机器学习等领域的应用。刘歆分别于2016年，2021年和2023年获得国家自然科学基金委优秀青年科学基金项目、杰出青年科学基金项目和科技部重点专项的资助。2024年获得中国工业与应用数学学会萧树铁应用数学奖。现担任MPC, JCM, APJOR等国内外期刊编委，《中国科学·数学》（中英文）青年编委，《计算数学》副主编；中国科学院青年创新促进会理事长；中国工业与应用数学学会秘书长；中国运筹学会常务理事；中国数学会计算数学分会常务理事。