A Unified Framework for Matrix Backpropagation

Computing matrix gradient has become a key aspect in modern signal processing/machine learning, with the recent use of matrix neural networks requiring matrix backpropagation. In this field, two main methods exist to calculate the gradient of matrix functions for symmetric positive definite (SPD) matrices, namely, the Daleckiǐ-Kreǐn/Bhatia formula and the Ionescu method. However, there appear to be a few errors. This brief aims to demonstrate each of these formulas in a self-contained and unified framework, to prove theoretically their equivalence, and to clarify inaccurate results of the literature. A numerical comparison of both methods is also provided in terms of computational speed and numerical stability to show the superiority of the Daleckiǐ-Kreǐn/Bhatia approach. We also extend the matrix gradient to the general case of diagonalizable matrices. Convincing results with the two backpropagation methods are shown on the EEG-based BCI competition dataset with the implementation of an SPDNet, yielding around 80% accuracy for one subject. Daleckiǐ-Kreǐn/Bhatia formula achieves an 8% time gain during training and handles degenerate cases.