方阵、宽阵和深阵的QR和LQ分解矩阵反向传播算法及其软件实现
QR and LQ Decomposition Matrix Backpropagation Algorithms for Square, Wide, and Deep -- Real or Complex -- Matrices and Their Software Implementation
摘要：
本文提出了矩阵$A_{m, n}$的QR分解的矩阵反向传播算法，这些矩阵可以是正方形（m = n）、宽（m < n）或深（m > n）的，秩为$k = min(m, n)$。此外，我们得到新的矩阵反向传播结果的枢轴（满秩）QR分解和深输入矩阵的LQ分解。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。其他用例，如图学习和网络压缩，在文章中列出。流行的深度学习框架（PyTorch，TensorFlow，MXNet）的软件实现包含了深度学习社区中通用的方法。此外，这篇文章帮助从业者理解矩阵反向传播方法作为更大的计算图的一部分。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
