Real order total variation with applications to the loss functions in learning schemes

Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a [Formula: see text]-order (an)-isotropic total variation semi-norms [Formula: see text], [Formula: see text], defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative [Formula: see text], of such loss functions.