Fast iterative regularization by reusing data
Name
10.1515_jiip-2023-0009.pdf
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3.45 MB
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Adobe PDF
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Author(s)
Cristian Vega
Cesare Molinari
Lorenzo Rosasco
Silvia Villa
Date Issued
October 27, 2023
Journal
Journal of Inverse and Ill-posed Problems
Publisher
Walter de Gruyter GmbH
Citation
Vega, Cristian, Molinari, Cesare, Rosasco, Lorenzo and Villa, Silvia. "Fast iterative regularization by reusing data" Journal of Inverse and Ill-posed Problems, vol. 32, no. 4, 2024, pp. 733-759.
Version
Final published version
Abstract
Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is neither smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to regularize inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set: we show that the linear equations determined by the data can be used more than once along the iterations. We discuss various approaches to reuse linear equations that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction. We confirm the efficiency of the proposed regularization approaches, comparing the results with state-of-the-art methods.
MIT Department
Center for Brains, Minds, and Machines
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1515/jiip-2023-0009
