  The maximal coding rate reduction (MCR$^2$) objective for learning structured
and compact deep representations is drawing increasing attention, especially
after its recent usage in the derivation of fully explainable and highly
effective deep network architectures. However, it lacks a complete theoretical
justification: only the properties of its global optima are known, and its
global landscape has not been studied. In this work, we give a complete
characterization of the properties of all its local and global optima, as well
as other types of critical points. Specifically, we show that each (local or
global) maximizer of the MCR$^2$ problem corresponds to a low-dimensional,
discriminative, and diverse representation, and furthermore, each critical
point of the objective is either a local maximizer or a strict saddle point.
Such a favorable landscape makes MCR$^2$ a natural choice of objective for
learning diverse and discriminative representations via first-order
optimization methods. To validate our theoretical findings, we conduct
extensive experiments on both synthetic and real data sets.
