Variational group-PCA for intrinsic dimensionality determination in fMRI data

Functional Magnetic Resonance Imaging (fMRI) is widely used to gain a better understanding of the human brain's functional organization. As fMRI data are high dimensional it is challenging to analyse using conventional methods thus low-rank approximations such as principal component analysis (PCA), and independent component analysis (ICA) is often applied as a preprocessing step before any additional analysis. Low-rank methods generally require that the rank or latent dimensionality is known beforehand. When this is not the case a range of plausible dimensionalities have to be tested and compared. Furthermore, in an fMRI-context it is not fully understood how information from multiple subjects should best be incorporated when applying dimensionality reduction. We propose a Bayesian group principal component analysis (Group-BPCA) model with an automatic relevance determination (ARD) prior to determine the number of active components supported by the data. All subjects share the same spatial maps (components), but the uncertainties on these maps as well as the noise is subject specific. We find an approximate solution using the mature variational Bayesian framework and develop a fast and scalable implementation using a graphical processing unit (GPU). We test the model on fMRI data from 29 healthy subjects performing a block-design fingertapping experiment. The model identified 10 active components. Neither variational Bayesian PCA on temporally concatenated data nor Group-BPCA, where uncertainties on the spatial maps are shared, leads to pruning components, but provide better generalization in two of three scenarios. We show that the right level of subject variability is highly dependent on the chosen validation scheme.