  We present a methodology for parallel acceleration of learning in the
presence of matrix orthogonality and unitarity constraints of interest in
several branches of machine learning. We show how an apparently sequential
elementary rotation parametrization can be restructured into blocks of
commutative operations using a well-known tool for coloring the edges of
complete graphs, in turn widely applied to schedule round-robin
(all-against-all) sports tournaments. The resulting decomposition admits an
algorithm to compute a fully-parametrized orthogonal matrix from its rotation
parameters in $O(n)$ sequential steps and one to compute the gradient of a
training loss with respect to its parameters in $O(n\log n)$ steps. We discuss
parametric restrictions of interest to generative modeling and present
promising performance results with a prototype GPU implementation.
