Recently developed applications in the field of machine learning and computational physics rely on automatic differentiation techniques, that require stable and efficient linear algebra gradient computations. This technical note provides a comprehensive and detailed discussion of the derivative of the truncated singular and eigenvalue decomposition. It summarizes previous work and builds on them with an extensive description of how to derive the relevant terms. A main focus is correctly expressing the derivative in terms of the truncated part, despite lacking knowledge of the full decomposition.

翻译：近年来，机器学习和计算物理领域新开发的应用依赖于自动微分技术，该技术要求稳定高效的线性代数梯度计算。本技术笔记全面详细地讨论了截断奇异值分解与特征值分解的导数。文章总结了前人工作，并在此基础上通过详尽描述如何推导相关项进行了拓展。主要重点在于：尽管缺乏完整分解的信息，仍能正确表达截断部分的导数。