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arXiv · 1612.05142

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

Elisa Davoli·Pan Liu
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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.

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Elisa Davoli, Pan Liu. 2016-12-15. One dimensional fractional order $TGV$: Gamma-convergence and bilevel training scheme. https://arxiv.org/abs/1612.05142

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