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 real-world 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.

翻译：在低秩优化中施加额外约束已引起越来越多的关注。然而，耦合约束的几何结构阻碍了已发展成熟的低秩结构，使得问题变得复杂。为此，我们提出了一种用于正交不变约束下有界秩矩阵优化的空间解耦框架。"空间解耦"体现在多个方面。我们证明了耦合约束的切锥是每个约束切锥的交集。此外，我们将相互交织的有界秩约束与正交不变约束解耦到两个空间中，从而导出了在光滑流形上的优化。只要额外约束的几何结构已知，在该流形上实现黎曼算法就变得轻而易举。此外，我们揭示了重构问题与原问题之间的等价性。在真实世界应用——球面数据拟合、图相似性度量、低秩半定规划、马尔可夫过程模型降阶、强化学习和深度学习——上的数值实验验证了所提框架的优越性。