Efficient Maximal Coding Rate Reduction by Variational Forms
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The principle of Maximal Coding Rate Reduction (MCR^2) has recently been proposed as a training objective for learning discriminative low-dimensional structures intrinsic to high-dimensional data to allow for more robust training than standard approaches, such as cross-entropy minimization. However, despite the advantages that have been shown for MCR^2 training, MCR^2 suffers from a significant computational cost due to the need to evaluate and differentiate a significant number of log-determinant terms that grows linearly with the number of classes. By taking advantage of variational forms of spectral functions of a matrix, we reformulate the MCR^2 objective to a form that can scale significantly without compromising training accuracy. Experiments in image classification demonstrate that our proposed formulation results in a significant speed up over optimizing the original MCR^2 objective directly and often results in higher quality learned representations. Further, our approach may be of independent interest in other models that require computation of log-determinant forms, such as in system identification or normalizing flow models.
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