1. Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-Order Convergence.
- Author
-
Luo, Yuetian, Huang, Wen, Li, Xudong, and Zhang, Anru
- Subjects
LEAST squares ,GAUSS-Newton method ,LOW-rank matrices ,MACHINE learning ,ALGORITHMS - Abstract
Solving Rank Constrained Least Squares via Recursive Importance Sketching In statistics and machine learning, we sometimes run into the rank-constrained least squares problems, for which we need to find the best low-rank fit between sets of data, such as trying to figure out what factors are affecting the data, filling in missing information, or finding connections between different sets of data. This paper introduces a new method for solving this problem called the recursive importance sketching algorithm (RISRO), in which the central idea is to break the problem down into smaller, easier parts using a unique technique called "recursive importance sketching." This new method is not only easy to use, but it is also very efficient and gives accurate results. We prove that RISRO converges in a local quadratic-linear and quadratic rate under some mild conditions. Simulation studies also demonstrate the superior performance of RISRO. In this paper, we propose a recursive importance sketching algorithm for rank-constrained least squares optimization (RISRO). The key step of RISRO is recursive importance sketching, a new sketching framework based on deterministically designed recursive projections, and it significantly differs from the randomized sketching in the literature. Several existing algorithms in the literature can be reinterpreted under this new sketching framework, and RISRO offers clear advantages over them. RISRO is easy to implement and computationally efficient, and the core procedure in each iteration is to solve a dimension-reduced least squares problem. We establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. We also discover a deep connection of RISRO to the Riemannian Gauss–Newton algorithm on fixed rank matrices. The effectiveness of RISRO is demonstrated in two applications in machine learning and statistics: low-rank matrix trace regression and phase retrieval. Simulation studies demonstrate the superior numerical performance of RISRO. Funding: Y. Luo and A. Zhang were partially supported by the National Science Foundation [Grant CAREER-2203741]. W. Huang was partially supported by the Fundamental Research Funds for the Central Universities [Grant 20720190060] and the National Natural Science Foundation of China [Grant 12001455]. X. Li was partially supported by the National Natural Science Foundation of China [Grants 62141407 and 12271107], the Chenguang Program by the Shanghai Education Development Foundation and Shanghai Municipal Education Commission [Grant 19CG02], and the Shanghai Science and Technology Program [Grant 21JC1400600]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2023.2445. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF