Quantifying low rank approximations of third order symmetric tensors

• 成果类型: Article
• 署名作者: Hu, Shenglong; Sun, Defeng; Toh, Kim-Chuan
• 署名单位: National University of Defense Technology - China; Hong Kong Polytechnic University; National University of Singapore
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-024-02165-1
• 发表日期: 2025
• 页码: 1119-1168
• 关键词: Augmented Lagrangian method; power method; decompositions; optimization; POLYNOMIALS

摘要

In this paper, we present a method to certify the approximation quality of a low rank tensor to a given third order symmetric tensor. Under mild assumptions, best low rank approximation is attained if a control parameter is zero or quantified quasi-optimal low rank approximation is obtained if the control parameter is positive. This is based on a primal-dual method for computing a low rank approximation for a given tensor. The certification is derived from the global optimality of the primal and dual problems, and is characterized by easily checkable relations between the primal and the dual solutions together with another rank condition. The theory is verified theoretically for orthogonally decomposable tensors as well as numerically through examples in the general case.