Correcting variance and polarity indeterminacies of extracted components by canonical polyadic decomposition

Background Back-projection has been used to correct the variance and polarity indeterminacies for the independent component analysis. The variance and polarity of the components are essential features of neuroscience studies.Objective This work extends the back-projection theory to canonical polyadic decomposition (CPD) for high-order tensors, aiming to correct the variance and polarity indeterminacies of the components extracted by CPD.Methods The tensor is reshaped into a matrix and decomposed using a suitable blind source separation algorithm. Subsequently, the coefficients are projected using back-projection theory, and other factor matrices are computed through a series of singular value decompositions of the back-projection matrix.Results By applying this method, the energy and polarity of each component are determined, effectively correcting the variance and polarity indeterminacies in CPD. The proposed method was validated using simulated tensor data and resting-state fMRI data.Conclusion Our proposed back-projection method for high-order tensors effectively corrects variance and polarity indeterminacies in CPD, offering a precise solution for calculating the energy and polarity required to extract meaningful features from neuroimaging data.