Efficient Maximal Coding Rate Reduction by Variational Forms

TL;DR

This paper introduces a variational reformulation of the Maximal Coding Rate Reduction (MCR$^2$) principle, significantly speeding up training and improving representation quality in image classification tasks.

Contribution

The authors propose a variational form of spectral functions to reformulate MCR$^2$, enabling scalable training without sacrificing accuracy.

Findings

Significant speed-up in MCR$^2$ training process.

Often higher quality learned representations.

Potential applications in other models requiring log-determinant computations.

Abstract

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.