Higher-Order Singular-Value Derivatives of Rectangular Real Matrices

• URL: http://arxiv.org/abs/2506.03764v3
• Date: Thu, 03 Jul 2025 11:36:31 GMT
• Title: Higher-Order Singular-Value Derivatives of Rectangular Real Matrices
• Authors: Róisín Luo, James McDermott, Colm O'Riordan,
• Abstract summary: We present a theoretical framework for deriving the general $n$-th order Fr'echet derivatives of singular values in real rectangular matrices.<n>We leverage reduced resolvent operators from Kato's perturbation analytic theory for self-adjoint operators.<n>Our framework equips researchers with a practical toolkit for higher-order spectral sensitivity studies in random matrix applications.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present a theoretical framework for deriving the general $n$-th order Fr\'echet 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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