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A theory of optimal convex regularization for low-dimensional recovery

Yann Traonmilin 1 Rémi Gribonval 2 Samuel Vaiter 3 
2 DANTE - Dynamic Networks : Temporal and Structural Capture Approach
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme, IXXI - Institut Rhône-Alpin des systèmes complexes
Abstract : 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 ℓ 1-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 example of sparsity in levels.
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https://hal.archives-ouvertes.fr/hal-03467123
Contributor : Yann Traonmilin Connect in order to contact the contributor
Submitted on : Monday, December 6, 2021 - 1:24:35 PM
Last modification on : Tuesday, October 25, 2022 - 4:20:29 PM

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  • HAL Id : hal-03467123, version 1
  • ARXIV : 2112.03540

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

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