QR and LQ Decomposition Matrix Backpropagation Algorithms for Square, Wide, and Deep -- Real or Complex -- Matrices and Their Software Implementation

TL;DR

This paper introduces novel matrix backpropagation algorithms for QR and LQ decompositions applicable to various matrix shapes, including square, wide, and deep matrices, with implementations in major deep learning frameworks.

Contribution

It develops new backpropagation algorithms for QR and LQ decompositions, including pivoted QR, and provides software implementations for deep learning frameworks.

Findings

Numerically stable and efficient QR-based least squares solutions

Backpropagation algorithms for pivoted QR and LQ decompositions

Software implementations in PyTorch, TensorFlow, MXNet

Abstract

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.