IEEE Trans Neural Syst Rehabil Eng. 2022;30:2630-2640. doi: 10.1109/TNSRE.2022.3198679. Epub 2022 Sep 19.
Complex-valued shift-invariant canonical polyadic decomposition (CPD) under a spatial phase sparsity constraint (pcsCPD) shows excellent separation performance when applied to band-pass filtered complex-valued multi-subject fMRI data. However, some useful information may also be eliminated when using a band-pass filter to suppress unwanted noise. As such, we propose an alternating rank- R and rank-1 least squares optimization to relax the CPD model. Based upon this optimization method, we present a novel constrained CPD algorithm with temporal shift-invariance and spatial sparsity and orthonormality constraints. More specifically, four steps are conducted until convergence for each iteration of the proposed algorithm: 1) use rank- R least-squares fit under spatial phase sparsity constraint to update shared spatial maps after phase de-ambiguity; 2) use orthonormality constraint to minimize the cross-talk between shared spatial maps; 3) update the aggregating mixing matrix using rank- R least-squares fit; 4) utilize shift-invariant rank-1 least-squares on a series of rank-1 matrices reconstructed by each column of the aggregating mixing matrix to update shared time courses, and subject-specific time delays and intensities. The experimental results of simulated and actual complex-valued fMRI data show that the proposed algorithm improves the estimates for task-related sensorimotor and auditory networks, compared to pcsCPD and tensorial spatial ICA. The proposed alternating rank- R and rank-1 least squares optimization is also flexible to improve CPD-related algorithm using alternating least squares.
在空间相位稀疏约束（pcsCPD）下的复值平移不变正则多元分解（CPD）应用于带通滤波的复值多体 fMRI 数据时，具有出色的分离性能。然而，当使用带通滤波器抑制不需要的噪声时，也可能会消除一些有用的信息。因此，我们提出了一种交替秩-R 和秩-1 最小二乘优化方法来放松 CPD 模型。基于这种优化方法，我们提出了一种新的具有时间平移不变性、空间稀疏性和正交性约束的约束 CPD 算法。具体来说，对于算法的每次迭代，我们进行四个步骤直到收敛：1）使用空间相位稀疏约束下的秩-R 最小二乘拟合来更新相位去模糊后的共享空间图谱；2）使用正交性约束来最小化共享空间图谱之间的串扰；3）使用秩-R 最小二乘拟合更新聚合混合矩阵；4）利用聚合混合矩阵的每一列重建的一系列秩-1 矩阵上的平移不变秩-1 最小二乘更新共享时间序列和个体特定的时滞和强度。模拟和实际复值 fMRI 数据的实验结果表明，与 pcsCPD 和张量空间 ICA 相比，所提出的算法提高了与任务相关的感觉运动和听觉网络的估计。所提出的交替秩-R 和秩-1 最小二乘优化也具有灵活性，可以使用交替最小二乘来改进 CPD 相关算法。
