Matrix factorizations in dual number algebra, a hypercomplex system, have been applied to kinematics, mechanisms, and other fields recently. We develop an approach to identify spatiotemporal patterns in the brain such as traveling waves using the singular value decomposition of dual matrices in this paper. Theoretically, we propose the compact dual singular value decomposition (CDSVD) of dual complex matrices with explicit expressions as well as a necessary and sufficient condition for its existence. Furthermore, based on the CDSVD, we report on the optimal solution to the best rank-$k$ approximation under a newly defined quasi-metric in the dual complex number system. The CDSVD is also related to the dual Moore-Penrose generalized inverse. Numerically, comparisons with other available algorithms are conducted, which indicate less computational costs of our proposed CDSVD. In addition, the infinitesimal part of the CDSVD can identify the true rank of the original matrix from the noise-added matrix, but the classical SVD cannot. Next, we employ experiments on simulated time-series data and a road monitoring video to demonstrate the beneficial effect of the infinitesimal parts of dual matrices in spatiotemporal pattern identification. Finally, we apply this approach to the large-scale brain fMRI data, identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function.

翻译：对偶数代数作为一种超复数系统，其矩阵分解方法已应用于运动学、机构学等领域。本文基于对偶矩阵的奇异值分解，提出一种识别脑部时空模式（如行波）的方法。理论上，我们给出了具有显式表达式的对偶复矩阵紧奇异值分解（CDSVD）及其存在的充要条件。此外，基于CDSVD，我们在对偶复数系中新定义的拟度量下，报告了最佳秩-$k$近似的最优解。CDSVD还与对偶Moore-Penrose广义逆相关。数值上，与其他现有算法的对比表明，我们提出的CDSVD计算成本更低。同时，CDSVD的无穷小部分可从含噪矩阵中识别原始矩阵的真实秩，而经典SVD无法做到。接着，我们通过模拟时间序列数据和道路监控视频实验，验证了对偶矩阵无穷小部分在时空模式识别中的有益作用。最后，我们将该方法应用于大规模脑功能磁共振成像（fMRI）数据，识别出三种行波，并进一步验证了分析结果与当前大脑皮层功能认知的一致性。