This paper introduces a tensor-based extension of Independent Component Analysis (ICA) for multi-subject functional magnetic resonance imaging (fMRI) analysis, called tensor-PICA. The method extends the single-session Probabilistic ICA (PICA) model to higher dimensions, enabling a three-way decomposition of fMRI data that captures temporal, spatial, and subject-dependent variations. The technique is compared with Parallel Factor Analysis (PARAFAC), which is a three-way array decomposition method. The tensor-PICA approach is shown to provide more accurate and robust results than PARAFAC, with reduced cross-talk between estimated sources, better computational efficiency, and improved interpretation of spatial, temporal, and subject/session modes. The method is applied to both simulated and real fMRI data, demonstrating its ability to extract meaningful activation maps, time courses, and session/subject modes. The tensor-PICA approach also provides a richer description of additional processes such as image artifacts or secondary activation patterns. The results show that tensor-PICA outperforms PARAFAC in terms of accuracy, robustness, and computational efficiency, particularly in cases where the data contains subject-specific sources or when there are small deviations from model assumptions. The method is also compared with mixed-effects GLMs, which are commonly used in group fMRI analysis. The tensor-PICA approach is shown to provide a more accurate and interpretable decomposition of multi-subject/multi-session fMRI data, making it a valuable tool for group fMRI studies. The paper also discusses the data preprocessing steps required for tensor-PICA, including voxel-wise detrending, de-meaning, co-registration, and temporal normalization. The results show that tensor-PICA is able to accurately estimate the spatial maps, time courses, and subject modes, with high correlation between the estimated and true sources. The method is shown to be robust to small deviations from model assumptions and to handle cases where the data contains subject-specific sources or where the temporal characteristics of the data differ between subjects. The paper concludes that tensor-PICA is a promising approach for multi-subject fMRI analysis, offering a more accurate and interpretable decomposition of the data compared to traditional methods.This paper introduces a tensor-based extension of Independent Component Analysis (ICA) for multi-subject functional magnetic resonance imaging (fMRI) analysis, called tensor-PICA. The method extends the single-session Probabilistic ICA (PICA) model to higher dimensions, enabling a three-way decomposition of fMRI data that captures temporal, spatial, and subject-dependent variations. The technique is compared with Parallel Factor Analysis (PARAFAC), which is a three-way array decomposition method. The tensor-PICA approach is shown to provide more accurate and robust results than PARAFAC, with reduced cross-talk between estimated sources, better computational efficiency, and improved interpretation of spatial, temporal, and subject/session modes. The method is applied to both simulated and real fMRI data, demonstrating its ability to extract meaningful activation maps, time courses, and session/subject modes. The tensor-PICA approach also provides a richer description of additional processes such as image artifacts or secondary activation patterns. The results show that tensor-PICA outperforms PARAFAC in terms of accuracy, robustness, and computational efficiency, particularly in cases where the data contains subject-specific sources or when there are small deviations from model assumptions. The method is also compared with mixed-effects GLMs, which are commonly used in group fMRI analysis. The tensor-PICA approach is shown to provide a more accurate and interpretable decomposition of multi-subject/multi-session fMRI data, making it a valuable tool for group fMRI studies. The paper also discusses the data preprocessing steps required for tensor-PICA, including voxel-wise detrending, de-meaning, co-registration, and temporal normalization. The results show that tensor-PICA is able to accurately estimate the spatial maps, time courses, and subject modes, with high correlation between the estimated and true sources. The method is shown to be robust to small deviations from model assumptions and to handle cases where the data contains subject-specific sources or where the temporal characteristics of the data differ between subjects. The paper concludes that tensor-PICA is a promising approach for multi-subject fMRI analysis, offering a more accurate and interpretable decomposition of the data compared to traditional methods.