提出矩形实矩阵奇异值高阶导数的闭式解法，解决传统方法难求的问题。
Higher-Order Singular-Value Derivatives of Rectangular Real Matrices
- 用自伴算子摄动理论构建矩阵块结构，捕捉非对称扰动。
- 推导出奇异值任意阶无穷小变化的闭式表达，首次给出二阶导数（海森矩阵）。
- 适用于深度学习对抗扰动等随机矩阵场景，提供实用分析工具。
我们提出一个理论框架，通过卡托的自伴算子解析摄动理论中的简化再生核算子，推导实矩形矩阵奇异值的n阶弗雷歇导数。传统矩阵分析方法难以获得奇异值高阶导数的闭式表达。为此，我们将实矩形矩阵视为有限维希尔伯特空间上的紧算子，并将其嵌入块自伴算子中以捕获非对称扰动。应用卡托的渐近本征值展开，我们得到无穷小n阶谱变的通用闭式表达。特别地，当n=2时，结合克罗内克积表示与矩阵约定，得出文献中未见的奇异值海森矩阵。本框架将抽象算子摄动理论与矩阵分析相衔接，为随机矩阵应用（如深度学习中的对抗扰动）中的高阶谱敏感性研究提供实用工具包。
原文摘要 · Abstract (English)
We present a theoretical framework for deriving the general $n$-th order Fréchet derivatives of singular values in real rectangular matrices, by leveraging reduced resolvent operators from Kato's analytic perturbation theory for self-adjoint operators. Deriving closed-form expressions for higher-order derivatives of singular values is notoriously challenging through standard matrix-analysis techniques. To overcome this, we treat a real rectangular matrix as a compact operator on a finite-dimensional Hilbert space, and embed the rectangular matrix into a block self-adjoint operator so that non-symmetric perturbations are captured. Applying Kato's asymptotic eigenvalue expansion to this construction, we obtain a general, closed-form expression for the infinitesimal $n$-th order spectral variations. Specializing to $n=2$ and deploying on a Kronecker-product representation with matrix convention yield the Hessian of a singular value, not found in literature. By bridging abstract operator-theoretic perturbation theory with matrices, our framework equips researchers with a practical toolkit for higher-order spectral sensitivity studies in random matrix applications (e.g., adversarial perturbation in deep learning).
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