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 $\epsilon_n$ and the time mesh-size $\tau_n$. We precisely characterize the asymptotic behaviour of $\epsilon_n$ and $\tau_n$ in order to ensure convergence to the considered minimizer. Finally, we apply our results to the setting of random graph models.

翻译：高阶正则化问题公式是机器学习、逆问题及图像/信号处理中广泛应用的框架。本文研究了带数据保真项的Sobolev $\mathrm{W}^{1,p}$半范数极小值求解的计算问题。我们提出了一种离散化方法，并证明了数值解与目标函数之间的收敛速率。该方法通过空间和时间维度对梯度流问题进行离散化。空间离散化采用了p-Laplacian算子的非局部近似，其收敛速率直接取决于局域化参数$\epsilon_n$和时间步长$\tau_n$。我们精确表征了$\epsilon_n$与$\tau_n$的渐近行为以确保收敛至所考虑的极小值。最后，我们将所得结论应用于随机图模型场景。