Empirical validation of task-related component analysis reformulation for computational complexity reduction

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The task-related component analysis (TRCA) is a data-driven method for extracting reproducible components across multiple data segments from multivariate data. TRCA has been proven effective in enhancing the signal-to-noise ratio of neuroimaging data in previous studies. However, its original form requires a computational cost of O(N2) to compute the sum of cross-covariance matrices, indicating that the computational time increases rapidly as the number of data segments N increases. This study aims to empirically validate that the reformulated form of TRCA can reduce its computational complexity. This study estimated that the reformulation can reduce the computational complexity from O(N2) to O(N) without distorting its outputs by reducing the number of matrix multiplications in computing the sum of cross-covariance matrices. A series of computer simulations using computer-generated synthetic, functional near-infrared spectroscopy (fNIRS), and electroencephalogram (EEG) data were conducted to verify the theoretical computation complexity and consistency of the outputs between the original and reformulated TRCA. The computation reduction was validated through an experimental comparison of the original and reformulated TRCA with synthetic and real data of various sizes (e.g., data segments, dimensions, and lengths). In addition, the two algorithms output almost exactly identical results with only floating-point errors. The reformulation would considerably benefit practical applications, especially when applying the TRCA to large-scale computations. The code is available at https://github.com/mnakanishi/TRCA-SSVEP.