Constrained CPD of Complex-Valued Multi-Subject fMRI Data via Alternating Rank-R and Rank-1 Least Squares

• Published in: IEEE transactions on neural systems and rehabilitation engineering Vol. 30; pp. 2630 - 2640
• Main Authors: Kuang, Li-Dan, Lin, Qiu-Hua, Gong, Xiao-Feng, Zhang, Jianming, Li, Wenjun, Li, Feng, Calhoun, Vince D.
• Format: Journal Article
• Language: English
• Published: United States IEEE 2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Band-pass filters; Bandpass filters; Brain - diagnostic imaging; Canonical polyadic decomposition (CPD); complex-valued fMRI data; Constraints; Crosstalk; Data models; Delay effects; Functional magnetic resonance imaging; Humans; Invariants; Iterative methods; Least squares; Least-Squares Analysis; Magnetic Resonance Imaging - methods; Matrix decomposition; Optimization; orthonormality; Sensorimotor system; shift-invariance; source phase sparsity; Sparsity; Task analysis; Tensors; Timing
• ISSN: 1534-4320; 1558-0210; 1558-0210
• DOI: 10.1109/TNSRE.2022.3198679

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-<inline-formula> <tex-math notation="LaTeX">{R} </tex-math></inline-formula> 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-<inline-formula> <tex-math notation="LaTeX">{R} </tex-math></inline-formula> 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-<inline-formula> <tex-math notation="LaTeX">{R} </tex-math></inline-formula> 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-<inline-formula> <tex-math notation="LaTeX">{R} </tex-math></inline-formula> and rank-1 least squares optimization is also flexible to improve CPD-related algorithm using alternating least squares.