A bilevel approach for parameter learning in inverse problems

Holler, Gernot; Kunisch, Karl; Barnard, Richard C.

doi:10.1088/1361-6420/aade77

Cited by 40 publications

(42 citation statements)

References 27 publications

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“…We derive necessary optimality conditions for the learning problem (BP). Here we essentially follow the discussion provided in [25,Section 5]. Throughout this section it is assumed that the function e describing the state constraint is well defined and at least once continuously F-differentiable on Y × H s (Ω).…”

Section: Optimality Conditionsmentioning

confidence: 99%

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Learning nonlocal regularization operators

Holler¹,

Kunisch²

2022

MCRF

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A learning approach for determining which operator from a class of nonlocal operators is optimal for the regularization of an inverse problem is investigated. The considered class of nonlocal operators is motivated by the use of squared fractional order Sobolev seminorms as regularization operators. First fundamental results from the theory of regularization with local operators are extended to the nonlocal case. Then a framework based on a bilevel optimization strategy is developed which allows to choose nonlocal regularization operators from a given class which i) are optimal with respect to a suitable performance measure on a training set, and ii) enjoy particularly favorable properties. Results from numerical experiments are also provided.

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Section: Optimality Conditionsmentioning

confidence: 99%

“…Learning strategies for choosing regularization parameters in the context of multi-penalty Tikhonov regularization are investigated e.g. in [28,14,11,25]. The problem of learning the discrepancy function is considered in [15].…”

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confidence: 99%

Learning nonlocal regularization operators

Holler¹,

Kunisch²

2022

MCRF

Self Cite

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“…For a given observation model U O , one would theoretically expect to retrieve some optimal parameterization Φ of regularization term U R such that the reconstruction error for the true state is truly minimized. This typically leads to considering the following bi-level optimization problem [2]…”

Section: Problem Statement and Related Workmentioning

confidence: 99%

“…We benefit from the considered end-to-end architecture to learn jointly all parameters, that is to say that we jointly learn the variational cost U and the associated solver so that we minimize the reconstruction error as targeted in (2). As such, we may learn a regularization term adapted to the available observation setting.…”

Section: Learning Schemementioning

confidence: 99%

End-to-End Learning of Variational Models and Solvers for the Resolution of Interpolation Problems

Fablet

Drumetz

Rousseau

2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Variational models are among the state-of-the-art formulations for the resolution of ill-posed inverse problems. Following recent advances in learning-based variational settings, we investigate the end-to-end learning of variational models, more precisely of the regularization term given some observation model, jointly to the associated solver, so that we can optimize the reconstruction performance. In the proposed end-to-end setting, both the variational cost and the gradient-based solver are stated as neural networks using automatic differentiation for the latter. We consider an application to inverse problems with incomplete datasets (image inpainting and multivariate time series interpolation). We experimentally illustrate that this framework can lead to a significant gain in terms of reconstruction performance, including w.r.t. the direct minimization of the variational formulation derived from the known generative model.

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“…In this paper we tackle the optimal placement problem using a bilevel learning approach [17,12,21]. In contrast to optimal experimental design strategies, our framework allows us to work with different quality measures and is not restricted to the A-, D-or E-optimal experimental design paradigms [31].…”

Section: Introductionmentioning

confidence: 99%

A bilevel learning approach for optimal observation placement in variational data assimilation

Castro¹,

Reyes²

2020

Inverse Problems

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In this paper we propose a bilevel optimization approach for the placement of space and time observations in variational data assimilation problems. Within the framework of supervised learning, we consider a bilevel problem where the lower-level task is the variational reconstruction of the initial condition of a semilinear system, and the upper-level problem solves the optimal placement with help of a sparsity inducing function. Due to the pointwise nature of the observations, an optimality system with regular Borel measures is obtained as necessary optimality condition for the lower-level problem. The latter is then considered as constraint for the upper-level instance, yielding an optimization problem constrained by a multi-state system with measures. We demonstrate existence of Lagrange multipliers and derive a necessary optimality system characterizing the optimal solutions of the bilevel problem. The numerical solution is carried out also on two levels. The lower-level problem is solved using a standard BFGS method, while the upper-level one is solved by means of a projected BFGS algorithm, based on the estimation of ǫ-active sets. Finally some numerical experiments are presented to illustrate the main features of our approach.

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A bilevel approach for parameter learning in inverse problems

Cited by 40 publications

References 27 publications

Learning nonlocal regularization operators

Learning nonlocal regularization operators

End-to-End Learning of Variational Models and Solvers for the Resolution of Interpolation Problems

A bilevel learning approach for optimal observation placement in variational data assimilation

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