Reduced rank regression for neural communication: a tutorial for neuroscientists

Reduced rank regression (RRR) is a statistical method for finding a low-dimensional linear mapping between a set of high-dimensional inputs and outputs. In recent years, RRR has found numerous applications in neuroscience, in particular for identifying "communication subspaces" governing the interactions between brain regions. This tutorial article seeks to provide an introduction to RRR and its mathematical foundations, with a particular emphasis on neural communication. We discuss RRR's relationship to alternate dimensionality reduction techniques such as singular value decomposition (SVD), principal components analysis (PCA), principal components regression (PCR), and canonical correlation analysis (CCA). We also derive important extensions to RRR, including ridge regularization and non-spherical noise. Finally, we introduce new metrics for quantifying communication strength as well as the alignment between communication axes and the principal modes of neural activity. By the end of this article, readers should have a clear understanding of RRR and the practical considerations involved in applying it to their own data.