Singular Value Decomposition of Dual Matrices and its Application to Traveling Wave Identification in the Brain

数学 奇异值分解 奇异值 基质(化学分析) 秩(图论) 应用数学 奇异谱分析 矩阵分解 算法 特征向量 组合数学 复合材料 材料科学 物理 量子力学
作者
Tong Wei,Weiyang Ding,Yimin Wei
出处
期刊:SIAM Journal on Matrix Analysis and Applications [Society for Industrial and Applied Mathematics]
卷期号:45 (1): 634-660 被引量:18
标识
DOI:10.1137/23m1556642
摘要

.Matrix factorizations in dual number algebra, a hypercomplex number system, have been applied to kinematics, spatial 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 (SVD) 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 functional magnetic resonance imaging data, identify three kinds of traveling waves, and further validate the consistency between our analytical results and the current knowledge of cerebral cortex function.Keywordsdual matricescompact dual SVDlow-rank approximationdual Moore–Penrose generalized inversetraveling wave identificationbrain dynamicstime-series data analysisMSC codes15A2315B3365F55
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