Monotone Lipschitz-Gradient Denoiser: Explainability of Operator Regularization Approaches Free From Lipschitz Constant Control
This paper addresses explainability of the operator-regularization approach under the use of monotone Lipschitz-gradient (MoL-Grad) denoiser - an operator that can be expressed as the Lipschitz continuous gradient of a differentiable convex function. We prove that an operator is a MoL-Grad denoiser...
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| Vydané v: | IEEE transactions on signal processing Ročník 73; s. 3378 - 3393 |
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| Jazyk: | English |
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| Abstract | This paper addresses explainability of the operator-regularization approach under the use of monotone Lipschitz-gradient (MoL-Grad) denoiser - an operator that can be expressed as the Lipschitz continuous gradient of a differentiable convex function. We prove that an operator is a MoL-Grad denoiser if and only if it is the "single-valued" proximity operator of a weakly convex function. An extension of Moreau's decomposition is also shown with respect to a weakly convex function and the conjugate of its convexified function. Under these arguments, two specific algorithms, the forward-backward splitting algorithm and the primal-dual splitting algorithm, are considered, both employing MoL-Grad denoisers. These algorithms generate a sequence of vectors converging weakly, under conditions, to a minimizer of a certain cost function which involves an "implicit regularizer" induced by the denoiser. Unlike the previous studies of operator regularization, our framework requires no control of the Lipschitz constant in learning the denoiser. The theoretical findings are supported by simulations. |
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| AbstractList | This paper addresses explainability of the operator-regularization approach under the use of monotone Lipschitz-gradient (MoL-Grad) denoiser - an operator that can be expressed as the Lipschitz continuous gradient of a differentiable convex function. We prove that an operator is a MoL-Grad denoiser if and only if it is the "single-valued" proximity operator of a weakly convex function. An extension of Moreau's decomposition is also shown with respect to a weakly convex function and the conjugate of its convexified function. Under these arguments, two specific algorithms, the forward-backward splitting algorithm and the primal-dual splitting algorithm, are considered, both employing MoL-Grad denoisers. These algorithms generate a sequence of vectors converging weakly, under conditions, to a minimizer of a certain cost function which involves an "implicit regularizer" induced by the denoiser. Unlike the previous studies of operator regularization, our framework requires no control of the Lipschitz constant in learning the denoiser. The theoretical findings are supported by simulations. |
| Author | Yukawa, Masahiro Yamada, Isao |
| Author_xml | – sequence: 1 givenname: Masahiro orcidid: 0000-0002-3709-275X surname: Yukawa fullname: Yukawa, Masahiro email: yukawa@elec.keio.ac.jp organization: Department of Electronics and Electrical Engineering, Keio University, Yokohama, Japan – sequence: 2 givenname: Isao orcidid: 0000-0002-6563-7526 surname: Yamada fullname: Yamada, Isao email: isao@sp.ict.e.titech.ac.jp organization: Department of Information and Communications Engineering, Institute of Science Tokyo, Japan |
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| References | ref13 Shimizu (ref57) 2024 ref12 ref15 ref59 ref14 ref58 ref53 ref52 ref11 ref55 Zeng (ref60) 2015 ref10 ref17 ref16 ref19 ref18 ref51 ref50 Hurault (ref69) 2022 ref46 ref45 ref48 ref42 ref41 ref43 Vincent (ref56) 2010; 11 ref49 ref8 ref7 ref9 ref4 ref3 ref6 ref5 ref40 Dinh (ref26) 1986; 129 ref35 ref34 Gao (ref54) 1997; 7 ref36 ref31 ref30 ref74 ref33 ref32 ref2 ref1 ref39 ref38 Moreau (ref47) 1962; 255 Yao (ref20) 2018; 18 ref71 ref70 ref73 ref72 Ryu (ref65) 2019; 97 ref24 ref23 ref67 Cohen (ref68) 2021; 34 ref25 ref64 ref63 ref22 ref66 ref21 ref28 ref27 ref29 Yukawa (ref44) 2023 ref62 Hurault (ref37) 2022; 162 ref61 |
| References_xml | – ident: ref51 doi: 10.1109/ICASSP48485.2024.10446368 – ident: ref1 doi: 10.1007/978-3-030-25939-6 – ident: ref6 doi: 10.1016/0898-1221(76)90003-1 – ident: ref5 doi: 10.1051/m2an/197509R200411 – ident: ref31 doi: 10.1109/TSP.2023.3263724 – ident: ref40 doi: 10.24033/bsmf.1625 – ident: ref21 doi: 10.1007/s10589-017-9954-1 – ident: ref52 doi: 10.1109/TSP.2015.2502551 – ident: ref15 doi: 10.1007/s00041-008-9045-x – ident: ref67 doi: 10.1137/16M1102884 – ident: ref34 doi: 10.1109/GlobalSIP.2013.6737048 – ident: ref49 doi: 10.1007/978-1-4419-9569-8_17 – ident: ref70 doi: 10.1109/TSP.2011.2107908 – ident: ref10 doi: 10.1007/s10107-018-1303-3 – volume: 129 start-page: 249 volume-title: Algorithms for solving a class of nonconvex optimization problems: Methods of subgradient year: 1986 ident: ref26 – ident: ref53 doi: 10.1109/LSP.2017.2710233 – ident: ref59 doi: 10.1109/TNNLS.2012.2200262 – volume: 18 start-page: 1 issue: 179 year: 2018 ident: ref20 article-title: Efficient learning with a family of nonconvex regularizers by redistributing nonconvexity publication-title: J. Mach. Learn. Res. – volume-title: Proc. IEICE Signal Process. Symp. year: 2023 ident: ref44 article-title: Cocoercive gradient operator of convex function and its associated weakly convex function: Generalized proximity operator for case of unique minimizer – ident: ref7 doi: 10.1007/s10851-010-0251-1 – ident: ref71 doi: 10.1109/LSP.2020.3006390 – ident: ref3 doi: 10.1137/0716071 – volume-title: Proc. IEICE Signal Process. Symp. year: 2024 ident: ref57 article-title: Implicit regularizer associated with tied-nonnegative-weight neural network: Plug-and-play for image restration and convergence to optimal point – ident: ref24 doi: 10.32614/cran.package.krls – volume: 34 start-page: 18152 volume-title: Proc. Adv. Neural Inf. Process. Syst. year: 2021 ident: ref68 article-title: It has potential: Gradient-driven denoisers for convergent solutions to inverse problems – ident: ref13 doi: 10.1198/016214501753382273 – year: 2015 ident: ref60 article-title: The ordered weighted $\ell_{1}$ Norm: Atomic formulation, projections, and algorithms – ident: ref74 doi: 10.1109/LSP.2014.2357681 – ident: ref41 doi: 10.1109/TSP.2016.2518989 – ident: ref27 doi: 10.7551/mitpress/7132.001.0001 – ident: ref36 doi: 10.1007/s10851-024-01181-2 – ident: ref32 doi: 10.1109/TSP.2023.3244082 – volume: 97 start-page: 5546 volume-title: Proc. Int. Conf. Mach. Learn. year: 2019 ident: ref65 article-title: Plug-and-play methods provably converge with properly trained denoisers – ident: ref25 doi: 10.1109/tit.2014.2312723 – ident: ref8 doi: 10.1007/s10957-012-0245-9 – ident: ref42 doi: 10.1109/TCI.2016.2599778 – ident: ref66 doi: 10.1109/TCI.2019.2893568 – ident: ref23 doi: 10.1137/151003714 – volume: 162 start-page: 9483 volume-title: Proc. ICML year: 2022 ident: ref37 article-title: Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization – ident: ref18 doi: 10.1109/TSP.2012.2212015 – ident: ref33 doi: 10.1109/ICASSP.2014.6853752 – volume: 255 start-page: 2897 year: 1962 ident: ref47 article-title: Fonctions convexes duales et points proximaux dans un espace hilbertien publication-title: C. R. Acad. Sci. Paris Ser. A Math. – ident: ref14 doi: 10.1088/0266-5611/24/3/035020 – ident: ref43 doi: 10.1109/ICCV.2017.198 – ident: ref50 doi: 10.1007/BF03007664 – ident: ref38 doi: 10.1137/23M1565243 – ident: ref64 doi: 10.1109/MSP.2022.3199595 – ident: ref62 doi: 10.1109/MSP.2019.2949470 – volume: 11 start-page: 3371 issue: 110 year: 2010 ident: ref56 article-title: Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion publication-title: J. Mach. Learn. Res. – ident: ref73 doi: 10.1137/17M1122451 – ident: ref4 doi: 10.1137/050626090 – ident: ref55 doi: 10.1080/10618600.1998.10474789 – ident: ref29 doi: 10.1109/TSP.2017.2711501 – ident: ref35 doi: 10.1137/20M1387961 – ident: ref46 doi: 10.1007/s10107-020-01500-6 – ident: ref58 doi: 10.1111/j.1541-0420.2007.00843.x – volume-title: Proc. ICLR year: 2022 ident: ref69 article-title: Gradient step denoiser for convergent plug-and-play – ident: ref11 doi: 10.1137/20M1379344 – ident: ref17 doi: 10.1214/09-AOS729 – ident: ref28 doi: 10.1109/83.784433 – volume: 7 start-page: 855 issue: 4 year: 1997 ident: ref54 article-title: WaveShrink with firm shrinkage publication-title: Statistica Sinica – ident: ref63 doi: 10.1109/MSP.2022.3207451 – ident: ref72 doi: 10.1109/TCI.2018.2880326 – ident: ref19 doi: 10.1109/TSP.2018.2824289 – ident: ref45 doi: 10.1007/s10851-020-00951-y – ident: ref48 doi: 10.1137/050626090 – ident: ref61 doi: 10.1109/OJSP.2025.3579646 – ident: ref30 doi: 10.1088/1361-6420/ab551e – ident: ref39 doi: 10.1007/978-3-319-48311-5 – ident: ref12 doi: 10.1214/07-SS014 – ident: ref16 doi: 10.1214/08-aos659 – ident: ref22 doi: 10.1109/LSP.2007.898300 – ident: ref2 doi: 10.1007/s11228-019-00526-z – ident: ref9 doi: 10.1007/s10444-011-9254-8 |
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| SubjectTerms | Algorithms Convergence Convex functions convex optimization Cost function Estimation Linear programming Lipschitz condition nonexpansive operator Operators (mathematics) proximity operator Regularization Signal processing Signal processing algorithms Splitting Sufficient conditions Training Vectors Weakly convex function |
| Title | Monotone Lipschitz-Gradient Denoiser: Explainability of Operator Regularization Approaches Free From Lipschitz Constant Control |
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