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Joint learning of variational representations and solvers for inverse problems with partially-observed data

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Document pages: 10 pages

Abstract: Designing appropriate variational regularization schemes is a crucial part ofsolving inverse problems, making them better-posed and guaranteeing that thesolution of the associated optimization problem satisfies desirable properties.Recently, learning-based strategies have appeared to be very efficient forsolving inverse problems, by learning direct inversion schemes or plug-and-playregularizers from available pairs of true states and observations. In thispaper, we go a step further and design an end-to-end framework allowing tolearn actual variational frameworks for inverse problems in such a supervisedsetting. The variational cost and the gradient-based solver are both stated asneural networks using automatic differentiation for the latter. We can jointlylearn both components to minimize the data reconstruction error on the truestates. This leads to a data-driven discovery of variational models. Weconsider an application to inverse problems with incomplete datasets (imageinpainting and multivariate time series interpolation). We experimentallyillustrate that this framework can lead to a significant gain in terms ofreconstruction performance, including w.r.t. the direct minimization of thevariational formulation derived from the known generative model.

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