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

Loss function are an essential part in modern data-driven approach, such as bi-level training scheme and machine learnings. In this paper we propose a loss function consisting of a r-order (an)-isotropic total variation semi-norms TVr, r∈ ℝ^+, defined via the Riemann-Liouville (R-L) 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 r, of such loss functions.