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Article Dans Une Revue SIAM Journal on Imaging Sciences Année : 2018

INEXACT HALF-QUADRATIC OPTIMIZATION FOR LINEAR INVERSE PROBLEMS

Résumé

We study the convergence of a generic half-quadratic algorithm for minimizing a wide class of objective functions that occur in inverse imaging problems; this algorithm amounts to solving a sequence of positive definite systems (the inner systems) and has the advantages of simplicity and versatility. Half-quadratic optimization has been meticulously studied, both theoretically and experimentally, but two difficulties remain: first, the practical solutions of the inner systems are generally approximate, which may hamper convergence, and, second, convergence to a stationary point of the objective is not guaranteed if the set of such points contains a continuum. We present new results that do not suffer from these limitations and hence extend our work in [SIAM J. Imaging Sci. 8 (2015), no. 3, 1752–1797]. We consider the inexact process in which the inner systems are solved to a fixed arbitrary accuracy defined in terms of the energy norm of the error. We show that this process converges to a stationary point of the objective under minimal conditions ubiquitous in regularized reconstruction and restoration. Our main results are based on the assumption that the objective has the Kurdyka-Lojasiewicz property, for which we provide constructing rules using the concept of tameness from the theory of o-minimal structures. We also propose an implementation using a truncated conjugate gradient method that controls the accuracy at negligible additional cost. Experiments on three different inverse problems show that the resulting algorithm performs well in various nonconvex scenarios and converges to solutions accurate to full machine precision.
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Dates et versions

hal-01584451 , version 1 (08-09-2017)
hal-01584451 , version 2 (09-05-2018)

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Citer

Marc C Robini, Feng Yang, Yuemin Zhu. INEXACT HALF-QUADRATIC OPTIMIZATION FOR LINEAR INVERSE PROBLEMS. SIAM Journal on Imaging Sciences, 2018, 11 (2), pp.1078 - 1133. ⟨10.1137/17M114635X⟩. ⟨hal-01584451v2⟩
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