  The suboptimal performance of wavelets with regard to the approximation of
multivariate data gave rise to new representation systems, specifically
designed for data with anisotropic features. Some prominent examples of these
are given by ridgelets, curvelets, and shearlets, to name a few.
  The great variety of such so-called directional systems motivated the search
for a common framework, which unites many under one roof and enables a
simultaneous analysis, for example with respect to approximation properties.
Building on the concept of parabolic molecules, the recently introduced
framework of $\alpha$-molecules does in fact include the previous mentioned
systems. Until now however it is confined to the bivariate setting, whereas
nowadays one often deals with higher dimensional data. This motivates the
extension of this unifying theory to dimensions larger than 2, put forward in
this work. In particular, we generalize the central result that the
cross-Gramian of any two systems of $\alpha$-molecules will to some extent be
localized.
  As an exemplary application, we investigate the sparse approximation of video
signals, which are instances of 3D data. The multivariate theory allows us to
derive almost optimal approximation rates for a large class of representation
systems.
