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Convex and Differentiable Formulation for Inverse Problems in Hilbert Spaces with Nonlinear Clipping Effects
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Summary :
We propose a useful formulation for ill-posed inverse problems in Hilbert spaces with nonlinear clipping effects. Ill-posed inverse problems are often formulated as optimization problems, and nonlinear clipping effects may cause nonconvexity or nondifferentiability of the objective functions in the case of commonly used regularized least squares. To overcome these difficulties, we present a tractable formulation in which the objective function is convex and differentiable with respect to optimization variables, on the basis of the Bregman divergence associated with the primitive function of the clipping function. By using this formulation in combination with the representer theorem, we need only to deal with a finite-dimensional, convex, and differentiable optimization problem, which can be solved by well-established algorithms. We also show two practical examples of inverse problems where our theory can be applied, estimation of band-limited signals and time-harmonic acoustic fields, and evaluate the validity of our theory by numerical simulations.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E104-A No.9 pp.1293-1303
- Publication Date
- 2021/09/01
- Publicized
- 2021/02/25
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.2021EAP1004
- Type of Manuscript
- PAPER
- Category
- Nonlinear Problems
Authors
Natsuki UENO
the University of Tokyo
Shoichi KOYAMA
the University of Tokyo
Hiroshi SARUWATARI
the University of Tokyo
Keyword
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Natsuki UENO, Shoichi KOYAMA, Hiroshi SARUWATARI, "Convex and Differentiable Formulation for Inverse Problems in Hilbert Spaces with Nonlinear Clipping Effects" in IEICE TRANSACTIONS on Fundamentals,
vol. E104-A, no. 9, pp. 1293-1303, September 2021, doi: 10.1587/transfun.2021EAP1004.
Abstract: We propose a useful formulation for ill-posed inverse problems in Hilbert spaces with nonlinear clipping effects. Ill-posed inverse problems are often formulated as optimization problems, and nonlinear clipping effects may cause nonconvexity or nondifferentiability of the objective functions in the case of commonly used regularized least squares. To overcome these difficulties, we present a tractable formulation in which the objective function is convex and differentiable with respect to optimization variables, on the basis of the Bregman divergence associated with the primitive function of the clipping function. By using this formulation in combination with the representer theorem, we need only to deal with a finite-dimensional, convex, and differentiable optimization problem, which can be solved by well-established algorithms. We also show two practical examples of inverse problems where our theory can be applied, estimation of band-limited signals and time-harmonic acoustic fields, and evaluate the validity of our theory by numerical simulations.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2021EAP1004/_p
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@ARTICLE{e104-a_9_1293,
author={Natsuki UENO, Shoichi KOYAMA, Hiroshi SARUWATARI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Convex and Differentiable Formulation for Inverse Problems in Hilbert Spaces with Nonlinear Clipping Effects},
year={2021},
volume={E104-A},
number={9},
pages={1293-1303},
abstract={We propose a useful formulation for ill-posed inverse problems in Hilbert spaces with nonlinear clipping effects. Ill-posed inverse problems are often formulated as optimization problems, and nonlinear clipping effects may cause nonconvexity or nondifferentiability of the objective functions in the case of commonly used regularized least squares. To overcome these difficulties, we present a tractable formulation in which the objective function is convex and differentiable with respect to optimization variables, on the basis of the Bregman divergence associated with the primitive function of the clipping function. By using this formulation in combination with the representer theorem, we need only to deal with a finite-dimensional, convex, and differentiable optimization problem, which can be solved by well-established algorithms. We also show two practical examples of inverse problems where our theory can be applied, estimation of band-limited signals and time-harmonic acoustic fields, and evaluate the validity of our theory by numerical simulations.},
keywords={},
doi={10.1587/transfun.2021EAP1004},
ISSN={1745-1337},
month={September},}
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TY - JOUR
TI - Convex and Differentiable Formulation for Inverse Problems in Hilbert Spaces with Nonlinear Clipping Effects
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1293
EP - 1303
AU - Natsuki UENO
AU - Shoichi KOYAMA
AU - Hiroshi SARUWATARI
PY - 2021
DO - 10.1587/transfun.2021EAP1004
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E104-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2021
AB - We propose a useful formulation for ill-posed inverse problems in Hilbert spaces with nonlinear clipping effects. Ill-posed inverse problems are often formulated as optimization problems, and nonlinear clipping effects may cause nonconvexity or nondifferentiability of the objective functions in the case of commonly used regularized least squares. To overcome these difficulties, we present a tractable formulation in which the objective function is convex and differentiable with respect to optimization variables, on the basis of the Bregman divergence associated with the primitive function of the clipping function. By using this formulation in combination with the representer theorem, we need only to deal with a finite-dimensional, convex, and differentiable optimization problem, which can be solved by well-established algorithms. We also show two practical examples of inverse problems where our theory can be applied, estimation of band-limited signals and time-harmonic acoustic fields, and evaluate the validity of our theory by numerical simulations.
ER -
English
