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

TL;DR

This paper introduces a new class of loss functions based on fractional-order total variation semi-norms, analyzing their key mathematical properties for potential applications in machine learning.

Contribution

It proposes a novel fractional-order total variation loss function and studies its theoretical properties like lower semi-continuity and compactness.

Findings

Establishment of lower semi-continuity of the fractional TV loss

Proof of compactness with respect to function and derivative order

Foundation for future applications in learning schemes

Abstract

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.