A space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints

• 成果类型: Article; Early Access
• 署名作者: Yang, Yan; Gao, Bin; Yuan, Ya-xiang
• 署名单位: Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02331-7
• 发表日期: 2026-02-04
• 关键词: Low-rank optimization; Orthogonal invariance; Tangent cone; Space decoupling; Riemannian Optimization; semidefinite programs; approximation; optimality; completion; algorithms; Similarity; points; cone

摘要

Imposing additional constraints on low-rank optimization has garnered growing interest. However, the geometry of coupled constraints hampers the well-developed low-rank structure and makes the problem intricate. To this end, we propose a space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints. The space-decoupling is reflected in several ways. We show that the tangent cone of coupled constraints is the intersection of tangent cones of each constraint. Moreover, we decouple the intertwined bounded-rank and orthogonally invariant constraints into two spaces, leading to optimization on a smooth manifold. Implementing Riemannian algorithms on this manifold is painless as long as the geometry of additional constraints is known. In addition, we unveil the equivalence between the reformulated problem and the original problem. Numerical experiments on fruitful applications-spherical data fitting, graph similarity measuring, low-rank SDP, model reduction of Markov processes, reinforcement learning, and deep learning-validate the superiority of the proposed framework.