Infinitesimal Higher-Order Spectral Variations in Rectangular Real Random Matrices

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).

@article{luo2025_2506.03764,
  title={ Higher-Order Singular-Value Derivatives of Rectangular Real Matrices },
  author={ Róisín Luo and James McDermott and Colm O'Riordan },
  journal={arXiv preprint arXiv:2506.03764},
  year={ 2025 }
}