Procrustes analysis for high-dimensional data

The Procrustes-based perturbation model \citep{Goodall} allows to minimize the Frobenius distance between matrices by similarity transformation. However, it suffers from non-identifiability, critical interpretation of the transformed matrices, and non-applicability 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 connectivity analysis since the ProMises model permits to incorporate topological brain information in the alignment's estimation process.