| Author: | Ma, Chiyu |
| Title: | A proximal generation based level set method with secant iterations for the least-squares constrained nuclear norm minimization |
| Advisors: | Sun, Defeng (AMA) |
| Degree: | Ph.D. |
| Year: | 2026 |
| Department: | Department of Applied Mathematics |
| Pages: | xviii, 102 pages : color illustrations |
| Language: | English |
| Abstract: | The least squares constrained nuclear norm minimization is a problem of practical significance but poses computational challenges. Its relevance spans numerous fields, including image processing, recommendation systems, system identification, classification, and clustering, among others. In this thesis, we propose a proximal generation based level set method with secant iterations to address this problem efficiently. Our approach combines a level set method with secant iterations and proximal generation techniques. The level set method reformulates the problem as a univariate nonlinear equation φ(λ) = ϱ derived from the regularized least squares problems. We employ the secant method for this equation and establish its fast convergence rates, which depend critically on two properties: the nonsingularity of ∂φ(λ) and (strong) semismoothness of φ(λ). We prove the nonsingularity of the Clarke generalized Jacobian for a broader class of univariate functions. For the semismooth analysis, we analyze the γ−order semismoothness through the semialgebraic property and derive a sufficient condition of the strong semismoothness through the implicit function theorem and the Lipschitz homeomorphism of the solution mapping. For the nuclear norm regularized subproblems within the level set method, we develop a proximal generation method that effectively exploits low-rank structures without compromising convergence. The method employs low-rank decomposition to reduce subproblems' computation. Proximal gradient steps are utilized to generate high-quality initial points for nonconvex problems and guarantee the global convergence simultaneously. Numerical experiments demonstrate the superior performance of our approach compared to existing methods. These experiments highlight our method's robustness, efficiency, and ability to handle large-scale problems, making it a valuable tool for applications in machine learning, data science, and signal processing where low-rank matrix optimization is prevalent. |
| Rights: | All rights reserved |
| Access: | open access |
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