  This article presents matrix backpropagation algorithms for the QR
decomposition of matrices $A_{m, n}$, that are either square (m = n), wide (m <
n), or deep (m > n), with rank $k = min(m, n)$. Furthermore, we derive novel
matrix backpropagation results for the pivoted (full-rank) QR decomposition and
for the LQ decomposition of deep input matrices. Differentiable QR
decomposition offers a numerically stable, computationally efficient method to
solve least squares problems frequently encountered in machine learning and
computer vision. Other use cases such as graph learning and network compression
are listed in the article. Software implementation across popular deep learning
frameworks (PyTorch, TensorFlow, MXNet) incorporate the methods for general use
within the deep learning community. Furthermore, this article aids the
practitioner in understanding the matrix backpropagation methodology as part of
larger computational graphs.
