A Global Geometric Analysis of Maximal Coding Rate Reduction
This provides a theoretical foundation for MCR^2, an objective used in explainable deep learning, addressing a gap in understanding its optimization landscape for researchers in representation learning.
The paper tackles the lack of theoretical justification for the maximal coding rate reduction (MCR^2) objective by characterizing all local and global optima and critical points, showing that maximizers yield low-dimensional, discriminative, and diverse representations, and critical points are either local maximizers or strict saddle points, making it suitable for first-order optimization.
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.