AN ALGORITHM SOLVING COMPRESSIVE SENSING PROBLEMS BASED ON MAXIMAL MONOTONE OPERATORS
Résumé
The need to solve $\ell^1$ regularized linear problems can be motivated by various compressive sensing and sparsity related techniques for data analysis and signal or image processing. These problems lead to nonsmooth convex optimization in high dimensions. Theoretical works predict a sharp phase transition for the exact recovery of compressive sensing problems. Our numerical experiments show that state-of-the-art algorithms are not effective enough to observe this phase transition accurately. This paper proposes a simple formalism that enables us to produce an algorithm that computes an $\ell^1$ minimizer under the constraints Au=b up to the machine precision. In addition, a numerical comparison with standard algorithms available in the literature is exhibited. The comparison shows that our algorithm compares advantageously with other state-of-the-art methods, both in terms of accuracy and efficiency. With our algorithm, the aforementioned phase transition is observed at high precision.
Mots clés
sparse solution recovery compressive sensing inverse scale space ℓ 1 minimization nonsmooth optimization maximal monotone operator phase transition AMS subject classifications. 34A60 49M29 90C06 90C25
sparse solution recovery
compressive sensing
inverse scale space
ℓ 1 minimization
nonsmooth optimization
maximal monotone operator
phase transition AMS subject classifications. 34A60
49M29
90C06
90C25
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