Higher-Order Singular-Value Derivatives of Rectangular Real Matrices
This work provides a practical toolkit for researchers studying higher-order spectral sensitivity in applications like adversarial perturbations in deep learning, though it is incremental as it builds on existing operator-theoretic perturbation theory.
The paper tackles the challenge of deriving closed-form expressions for higher-order derivatives of singular values in real rectangular matrices, achieving a general formula for the n-th order Fréchet derivatives and specifically providing the Hessian for the second-order case, which was previously unavailable in literature.
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).