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A theory of optimal convex regularization for low-dimensional recovery

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Résumé

We consider the problem of recovering elements of a low-dimensional model from under-determined linear measurements. To perform recovery, we consider the minimization of a convex regularizer subject to a data fit constraint. Given a model, we ask ourselves what is the ``best'' convex regularizer to perform its recovery. To answer this question, we define an optimal regularizer as a function that maximizes a compliance measure with respect to the model. We introduce and study several notions of compliance. We give analytical expressions for compliance measures based on the best-known recovery guarantees with the restricted isometry property. These expressions permit to show the optimality of the ℓ1-norm for sparse recovery and of the nuclear norm for low-rank matrix recovery for these compliance measures. We also investigate the construction of an optimal convex regularizer using the examples of sparsity in levels and of sparse plus low-rank models.
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Dates et versions

hal-03467123 , version 1 (06-12-2021)
hal-03467123 , version 2 (12-12-2022)

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Yann Traonmilin, Rémi Gribonval, Samuel Vaiter. A theory of optimal convex regularization for low-dimensional recovery. 2022. ⟨hal-03467123v2⟩
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