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The principle of Maximal Coding Rate Reduction (MCR$^2$) has recently beenproposed as a training objective for learning discriminative low-dimensionalstructures intrinsic to high-dimensional data to allow for more robust trainingthan standard approaches, such as cross-entropy minimization. However, despitethe advantages that have been shown for MCR$^2$ training, MCR$^2$ suffers froma significant computational cost due to the need to evaluate and differentiatea significant number of log-determinant terms that grows linearly with thenumber of classes. By taking advantage of variational forms of spectralfunctions of a matrix, we reformulate the MCR$^2$ objective to a form that canscale significantly without compromising training accuracy. Experiments inimage classification demonstrate that our proposed formulation results in asignificant speed up over optimizing the original MCR$^2$ objective directlyand often results in higher quality learned representations. Further, ourapproach may be of independent interest in other models that requirecomputation of log-determinant forms, such as in system identification ornormalizing flow models.

Efficient Maximal Coding Rate Reduction by Variational Forms