Optimizing pcsCPD with Alternating Rank-R and Rank-1 Least Squares: Application to Complex-Valued Multi-subject fMRI Data

Li-Dan Kuang, Wenjun Li, Yan Gui

Abstract: Complex-valued shift-invariant canonical polyadic decomposition (CPD) under a spatial phase sparsity constraint (pcsCPD) showed satisfying separation performance of decomposing three-way multi-subject fMRI data into shared spatial maps (SMs), shared time courses (TCs), time delays and subject-specific intensities. However, pcsCPD exploits alternating least squares (ALS) updating rule, which converges slowly and requires data strictly conforming to the shift-invariant CPD model. As the lower rank approximation can relax the CPD model, we propose to improve pcsCPD with rank-R and rank-1 ALS to further relax shift-invariant CPD model. This proposed method firstly updates shared SMs and the aggregating mixing matrix which contains the information of shared TCs, time delays and subject-specific intensities using the rank-R ALS. The shared SMs then are second updated by exploiting the phase sparsity constraint. We further update the shared TCs, time delays and subject-specific intensities of each component by the rank-1 ALS on the matrix constructed by each column of the aggregating mixing matrix, for each iteration until convergence. Experiment results from simulated and experimental fMRI data demonstrate that the proposed method achieves better separation performance than pcsCPD and widely-used tensor-based spatial independent component analysis, suggesting the efficacy of relaxing the shift-invariant CPD modelling of multi-subject fMRI data.