  The Procrustes-based perturbation model (Goodall, 1991) allows minimization
of the Frobenius distance between matrices by similarity transformation.
However, it suffers from non-identifiability, critical interpretation of the
transformed matrices, and inapplicability in high-dimensional data. We provide
an extension of the perturbation model focused on the high-dimensional data
framework, called the ProMises (Procrustes von Mises-Fisher) model. The
ill-posed and interpretability problems are solved by imposing a proper prior
distribution for the orthogonal matrix parameter (i.e., the von Mises-Fisher
distribution) which is a conjugate prior, resulting in a fast estimation
process. Furthermore, we present the Efficient ProMises model for the
high-dimensional framework, useful in neuroimaging, where the problem has much
more than three dimensions. We found a great improvement in functional magnetic
resonance imaging (fMRI) connectivity analysis because the ProMises model
permits incorporation of topological brain information in the alignment's
estimation process.
