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A novel class of semi-norms, generalising the notion of the isotropic totalvariation $TV_{2}$ and the an-isotropic total variation $TV_{1}$ is introduced.A supervised learning method via bilevel optimisation is proposed for thecomputation of optimal parameters for this class of regularizers. Existence ofsolutions to the bilevel optimisation approach is proven. Moreover, afinite-dimensional approximation scheme for the bilevel optimisation approachis introduced that can numerically compute a global optimizer to any givenaccuracy.

Learning optimal orders of the underlying Euclidean norm in total variation image denoising