Discrete-to-Continuum Rates of Convergence for $p$-Laplacian Regularization

Abstract

Higher-order regularization problem formulations are popular frameworks used in machine learning, inverse problems and image/signal processing. In this paper, we consider the computational problem of finding the minimizer of the Sobolev $\mathrm{W}^{1,p}$ semi-norm with a data-fidelity term. We propose a discretization procedure and prove convergence rates between our numerical solution and the target function. Our approach consists of discretizing an appropriate gradient flow problem in space and time. The space discretization is a nonlocal approximation of the p-Laplacian operator and our rates directly depend on the localization parameter $ε_n$ and the time mesh-size $τ_n$. We precisely characterize the asymptotic behaviour of $ε_n$ and $τ_n$ in order to ensure convergence to the considered minimizer. Finally, we apply our results to the setting of random graph models.