One dimensional fractional order $TGV$: Gamma-convergence and bilevel training scheme

TL;DR

This paper introduces fractional order TGV seminorms in 1D, proves their properties, and develops a bilevel training scheme with convergence analysis and numerical insights.

Contribution

It generalizes integer order TGV to fractional orders, establishes a bilevel optimization framework, and proves solution existence via Gamma-convergence.

Findings

Fractional TGV seminorms are intermediate between integer orders.

Existence of solutions to the bilevel scheme is established.

Numerical analysis of the cost function landscape is provided.

Abstract

New fractional $r$-order seminorms, $TGV^r$, $r\in \mathbb R$, $r\geq 1$, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order $TGV^k$-seminorms, $k\in\mathbb{N}$. The fractional $r$-order $TGV^r$-seminorms are shown to be intermediate between the integer order $TGV^k$-seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect to parameters and order of derivation is performed. Existence of solutions to the bilevel training scheme is proved by $\Gamma$-convergence. Finally, the numerical landscape of the cost function associated to the bilevel training scheme is discussed for two numerical examples.