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The decomposition of multi-subject fMRI data using rank- (L,L,1,1) block term decomposition (BTD) can preserve higher-way data structure and is more robust to noise effects by decomposing shared spatial maps (SMs) into a product of two rank-L loading matrices. However, since the number of whole-brain voxels is very large and rank L is larger than 1, the rank-(L,L,1,1) BTD requires high computation and memory. Therefore, we propose an accelerated rank- (L,L,1,1) BTD algorithm based upon the method of alternating least squares (ALS). We speed up updates of loading matrices by reducing fMRI data into subspaces, and add an orthonormality constraint on shared SMs to improve the performance. Moreover, we evaluate the rank-L effect on the proposed method for actual task-related fMRI data. The proposed method shows better performance when L=35. Meanwhile, experimental comparison results verify that the proposed method largely reduced (17.36 times) computation time compared to ALS while also providing satisfying separation performance. 

Brain networks extracted by independent component analysis (ICA) from magnitude‐only fMRI data are usually denoised using various amplitude‐based thresholds. By contrast, spatial source phase (SSP) or the phase information of ICA brain networks extracted from complex‐valued fMRI data, has provided a simple yet effective way to perform the denoising using a fixed phase change. In this work, we extend the approach to magnitude‐only fMRI data to avoid testing various amplitude thresholds for denoising magnitude maps extracted by ICA, as most studies do not save the complex‐valued data. The main idea is to generate a mathematical SSP map for a magnitude map using a mapping framework, and the mapping framework is built using complex‐valued fMRI data with a known SSP map. Here we leverage the fact that the phase map derived from phase fMRI data has similar phase information to the SSP map. After verifying the use of the magnitude data of complex‐valued fMRI, this framework is generalized to work with magnitude‐only data, allowing use of our approach even without the availability of the corresponding phase fMRI datasets. We test the proposed method using both simulated and experimental fMRI data including complex‐valued data from University of New Mexico and magnitude‐only data from Human Connectome Project. The results provide evidence that the mathematical SSP denoising with a fixed phase change is effective for denoising spatial maps from magnitude‐only fMRI data in terms of retaining more BOLD‐related activity and fewer unwanted voxels, compared with amplitude‐based thresholding. The proposed method provides a unified and efficient SSP approach to denoise ICA brain networks in fMRI data.