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. 2020 Jan 20;10:680. doi: 10.1038/s41598-020-57513-w

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© The Author(s) 2020, corrected publication 2022

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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The concepts of tensor and matrix decomposition. (a) Tensor decomposition and CP decomposition, our focus in the current study. For decomposition, we construct a 3-dimensional array of data consisting of S columns, T rows, and K slices. After decomposition, we obtain spatial modules, as shown by the bar graph in the blue frame; temporal modules, as shown by the line plot in the green frame; and the task-dependent modulations of these modules, shown as circular dots in the red frame. (b) Matrix decomposition. To analyze K task datasets simultaneously, we need to establish an S × (T × K) matrix. After applying matrix decomposition, we obtain the spatial modules and the task-dependent modulations of the temporal modules.