提出截断奇异值与特征分解的稳定求导方法，解决机器学习中的梯度计算难题。
Derivative of the truncated singular value and eigen decomposition
- 基于截断部分重构导数表达式，无需完整分解信息
- 推导出可直接用于自动微分的解析导数公式
- 适合需要高效线性代数梯度的机器学习与物理模拟场景
近年来，机器学习和计算物理领域的新应用依赖于自动微分技术，这要求线性代数梯度计算具备稳定性和高效性。本文系统详尽地讨论了截断奇异值分解（truncated SVD）和特征值分解的导数问题。总结了已有研究成果，并在此基础上深入阐述相关项的推导过程。重点在于如何在未知完整分解的情况下，正确用截断部分表示导数，为实际应用提供理论支持与实现路径。
原文摘要 · Abstract (English)
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.
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