The J-orthogonal matrix, also referred to as the hyperbolic orthogonal matrix, is a class of special orthogonal matrix in hyperbolic space, notable for its advantageous properties. These matrices are integral to optimization under J-orthogonal constraints, which have widespread applications in statistical learning and data science. However, addressing these problems is generally challenging due to their non-convex nature and the computational intensity of the constraints. Currently, algorithms for tackling these challenges are limited. This paper introduces JOBCD, a novel Block Coordinate Descent method designed to address optimizations with J-orthogonality constraints. We explore two specific variants of JOBCD: one based on a Gauss-Seidel strategy (GS-JOBCD), the other on a variance-reduced and Jacobi strategy (VR-J-JOBCD). Notably, leveraging the parallel framework of a Jacobi strategy, VR-J-JOBCD integrates variance reduction techniques to decrease oracle complexity in the minimization of finite-sum functions. For both GS-JOBCD and VR-J-JOBCD, we establish the oracle complexity under mild conditions and strong limit-point convergence results under the Kurdyka-Lojasiewicz inequality. To demonstrate the effectiveness of our method, we conduct experiments on hyperbolic eigenvalue problems, hyperbolic structural probe problems, and the ultrahyperbolic knowledge graph embedding problem. Extensive experiments using both real-world and synthetic data demonstrate that JOBCD consistently outperforms state-of-the-art solutions, by large margins.

翻译：J-正交矩阵，亦称双曲正交矩阵，是双曲空间中一类特殊的正交矩阵，以其优越的性质而著称。这类矩阵在J-正交约束下的优化问题中不可或缺，在统计学习和数据科学中具有广泛的应用。然而，由于问题的非凸性以及约束本身的计算复杂性，解决这类问题通常具有挑战性。目前，应对这些挑战的算法仍较为有限。本文提出了JOBCD，一种新颖的块坐标下降方法，旨在处理具有J-正交约束的优化问题。我们探讨了JOBCD的两种具体变体：一种基于高斯-赛德尔策略（GS-JOBCD），另一种基于方差缩减与雅可比策略（VR-J-JOBCD）。值得注意的是，VR-J-JOBCD利用雅可比策略的并行框架，集成了方差缩减技术，以降低有限和函数最小化的预言机复杂度。对于GS-JOBCD和VR-J-JOBCD，我们在温和条件下建立了其预言机复杂度，并在Kurdyka-Lojasiewicz不等式下证明了强极限点收敛性。为了验证我们方法的有效性，我们在双曲特征值问题、双曲结构探针问题以及超双曲知识图谱嵌入问题上进行了实验。使用真实世界数据和合成数据的大量实验表明，JOBCD始终以显著优势超越现有最先进的解决方案。