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Loss function are an essential part in modern data-driven approach, such asbi-level training scheme and machine learnings. In this paper we propose a lossfunction consisting of a $r$-order (an)-isotropic total variation semi-norms$TV^r$, $r\in \mathbb{R}^+$, defined via the Riemann-Liouville (R-L) fractionalderivative. We focus on studying key theoretical properties, such as the lowersemi-continuity and compactness with respect to both the function and the orderof derivative $r$, of such loss functions.

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