We consider the block coordinate descent methods of Gauss-Seidel type with proximal regularization (BCD-PR), which is a classical method of minimizing general nonconvex objectives under constraints that has a wide range of practical applications. We theoretically establish the worst-case complexity bound for this algorithm. Namely, we show that for general nonconvex smooth objectives with block-wise constraints, the classical BCD-PR algorithm converges to an epsilon-stationary point within O(1/epsilon) iterations. Under a mild condition, this result still holds even if the algorithm is executed inexactly in each step. As an application, we propose a provable and efficient algorithm for `Wasserstein CP-dictionary learning', which seeks a set of elementary probability distributions that can well-approximate a given set of d-dimensional joint probability distributions. Our algorithm is a version of BCD-PR that operates in the dual space, where the primal problem is regularized both entropically and proximally.
University of California, Los Angeles（加州大学洛杉矶分校）；University of South Carolina（南卡罗来纳大学）；Max Planck Institute for Mathematics in Sciences（马克斯·普朗克数学科学研究所）
Zhiyuan College, Shanghai Jiao Tong University, Shanghai 200240, China（上海交通大学紫阳学院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen, Guangdong, China（香港中文大学（深圳）数据科学学院）；McCombs School of Business, The University of Texas at Austin, Austin, TX, USA（德克萨斯大学奥斯汀分校麦克拉姆商学院）
Department of Mathematics, University of California, Los Angeles, CA 90095（加州大学洛杉矶分校数学系）；Department of Mathematics, University of South Carolina, Columbia, SC 29208（南卡罗来纳大学数学系）
Sparse Transformer Architectures via Regularized Wasserstein Proximal Operator with $L_1$ Prior
Department of Mathematics, University of California, Los Angeles, CA, USA（加州大学洛杉矶分校数学系）；Department of Mathematics, University of South Carolina, Columbia, SC, USA（南卡罗来纳大学数学系）
Linear convergence of proximal descent schemes on the Wasserstein space
Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan（日本东京RIKEN先进智能研究中心）；School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, UK（赫瑞-瓦特大学数学与计算机科学学院）；Maxwell Institute for Mathematical Sciences, Edinburgh, UK（爱丁堡数学科学麦克斯韦研究所）；School of Mathematics, University of Edinburgh, UK（爱丁堡大学数学学院）；School of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK（爱丁堡大学数学学院、英国艾伦·图灵研究所）
Convergence of flow-based generative models via proximal gradient descent in Wasserstein space
Department of Mathematics, Duke University（杜克大学数学系）；Department of Physics and Department of Chemistry, Duke University（杜克大学物理系和化学系）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
Wasserstein proximal operators describe score-based generative models and resolve memorization
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of South Carolina, Columbia（南卡罗来纳大学哥伦比亚分校）
Fast and Provably Convergent Algorithms for Gromov-Wasserstein in Graph Data
Stanford University（斯坦福大学）；Hong Kong University of Science and Technology（香港科技大学）；The Chinese University of Hong Kong（香港中文大学）；Shanghai University of Finance and Economics（上海财经大学）
A stochastic smoothing framework for nonconvex-nonconcave minEmax problems with applications to Wasserstein distributionally robust optimization
Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China（应用数学系，香港理工大学）；Institute for Math & AI, Wuhan (IMAI), Wuhan university, China（数学与人工智能研究院，武汉大学）；Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY, USA（数学科学系，伦斯勒理工学院）
Enhancing Distributional Robustness in Principal Component Analysis by Wasserstein Distances
The Hong Kong Polytechnic University（香港理工大学）；Chinese Academy of Sciences（中国科学院）；Academy of Mathematics and Systems Science（数学系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
Distributionally Time-Varying Online Stochastic Optimization under Polyak-Łojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning
The Australian National University（澳大利亚国立大学）；The University of Melbourne（墨尔本大学）
