Real order total variation with applications to the loss functions in learning schemes
This work addresses the need for more flexible loss functions in data-driven methods, but it appears incremental as it extends existing total variation concepts to fractional orders without clear application results.
The paper tackles the problem of designing loss functions for machine learning by introducing a new family based on real-order total variation using fractional derivatives, and establishes key theoretical properties like lower semi-continuity and compactness.
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 $TV^r$, $r\in \mathbb{R}^+$, 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.