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

arXiv: https://arxiv.org/abs/2009.10071 · HTML: https://mlsanity.com/paper/2009.10071/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2020.

## Built on by (most cited first, 3 of 3)

- [Diagonalization without Diagonalization: A Direct Optimization Approach for Solid-State Density Functional Theory](https://mlsanity.com/paper/2411.05033/) (2024)
- [Learning Molecular Chirality via Chiral Determinant Kernels](https://mlsanity.com/paper/2602.07415/) (2026)
- [Exact and general decoupled solutions of the LMC Multitask Gaussian Process model](https://mlsanity.com/paper/2310.12032/) (2023)

## Abstract (arXiv)

This article presents matrix backpropagation algorithms for the QR decomposition of matrices Am, 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.