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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.03764 |
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| _version_ | 1866909901317472256 |
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| author | Luo, Róisín McDermott, James O'Riordan, Colm |
| author_facet | Luo, Róisín McDermott, James O'Riordan, Colm |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03764 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher-Order Singular-Value Derivatives of Rectangular Real Matrices Luo, Róisín McDermott, James O'Riordan, Colm Machine Learning 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). |
| title | Higher-Order Singular-Value Derivatives of Rectangular Real Matrices |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2506.03764 |