基于结构化因子的大规模矩阵与张量分解计算：一个统一的非凸优化视角
Computing Large-Scale Matrix and Tensor Decomposition with Structured Factors: A Unified Nonconvex Optimization Perspective
AI总结：
本文采用自上而下的非凸优化视角，从通用优化理论出发讲解结构化矩阵与张量分解的算法设计，重点介绍了随机草图、自适应步长随机优化等前沿方法，兼具教育价值并旨在推动该领域研究。
摘要：
本文旨在为结构化矩阵与张量分解的计算层面提供全面教程。与现有教程主要聚焦于少数问题（如非负约束或稀疏约束分解）的算法流程不同，我们采用自上而下的方法：从通用优化理论（包括非精确与加速块坐标下降、随机优化以及高斯-牛顿方法）入手，该理论可覆盖具有工程应用价值的各类约束与正则项的广泛分解问题。随后，我们深入底层，展示在上述原理指导下的具体算法设计。我们特别关注结构化张量与矩阵分解领域的最新算法进展，例如基于随机草图与自适应步长的随机优化，以及利用结构特性的二阶算法——这些均为当前最先进的方法，但相较于基于块坐标下降（BCD）的方法，现有文献对其的探讨要少得多。我们期望本文能在结构化分解领域发挥教育价值，并希望推动更多研究投入这一重要且充满活力的方向。
英文摘要：
The proposed article aims at offering a comprehensive tutorial for the computational aspects of structured matrix and tensor factorization. Unlike existing tutorials that mainly focus on {\it algorithmic procedures} for a small set of problems, e.g., nonnegativity or sparsity-constrained factorization, we take a {\it top-down} approach: we start with general optimization theory (e.g., inexact and accelerated block coordinate descent, stochastic optimization, and Gauss-Newton methods) that covers a wide range of factorization problems with diverse constraints and regularization terms of engineering interest. Then, we go `under the hood' to showcase specific algorithm design under these introduced principles. We pay a particular attention to recent algorithmic developments in structured tensor and matrix factorization (e.g., random sketching and adaptive step size based stochastic optimization and structure-exploiting second-order algorithms), which are the state of the art---yet much less touched upon in the literature compared to {\it block coordinate descent} (BCD)-based methods. We expect that the article to have an educational values in the field of structured factorization and hope to stimulate more research in this important and exciting direction.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
