Spectral-transport stability and benign overfitting for minimum norm interpolation

Abstract

Benign overfitting describes the ability of minimum norm interpolating estimators to generalize despite fitting noisy data exactly. Existing characterizations depend on delicate spectral functionals of the population covariance operator, namely the effective ranks of its eigenvalue tail. We study the stability of these characterizations when the covariance spectrum is perturbed, and we quantify perturbations with the Wasserstein distance between spectral measures, a viewpoint we call spectral transport. We prove that eigenvalue tail sums, tail second moments, and the two effective ranks that govern benign overfitting are Lipschitz stable with respect to the spectral-transport distance, with explicit constants driven by an eigenvalue gap. As consequences we obtain three results: a risk transfer theorem for the minimum norm interpolator under aligned spectral perturbations, a stability theorem showing that the benign overfitting classification is preserved under vanishing spectral-transport perturbations, and an empirical certification result in which sample covariance spectra certify benignity through operator norm concentration. The framework connects benign overfitting to harmonic analysis constructions such as diffusion maps and scattering representations, where covariance spectra are perturbed by deformations of the data representation, and to linearized optimal transport, where Wasserstein geometry is the natural metric on perturbations. Numerical experiments with three spectral families confirm the theory: the ordering of the effective rank indices predicts the ordering of the empirical excess risks, and the observed risk change scales at a near Lipschitz rate in the Wasserstein distance between spectra.