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. 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, and hopefully, leads to new lines of research.

中文翻译：

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方、宽、深矩阵的 QR 和 LQ 分解矩阵反向传播算法及其软件实现

本文介绍了矩阵 $A_{m, n}$ 的 QR 分解的矩阵反向传播算法，这些矩阵是正方形 (m = n)、宽 (m < n) 或深 (m > n)，秩为 $k = min(m, n)$。此外，我们为枢轴（满秩）QR 分解和深度输入矩阵的 LQ 分解推导出新的矩阵反向传播结果。可微 QR 分解提供了一种数值稳定、计算效率高的方法来解决机器学习和计算机视觉中经常遇到的最小二乘问题。跨流行的深度学习框架（PyTorch、TensorFlow、MXNet）的软件实现结合了深度学习社区中通用的方法。此外，