Pub Date : 2024-08-11eCollection Date: 2024-01-01DOI: 10.1080/01630563.2024.2384849
Mehrsa Pourya, Sebastian Neumayer, Michael Unser
We propose a regularization scheme for image reconstruction that leverages the power of deep learning while hinging on classic sparsity-promoting models. Many deep-learning-based models are hard to interpret and cumbersome to analyze theoretically. In contrast, our scheme is interpretable because it corresponds to the minimization of a series of convex problems. For each problem in the series, a mask is generated based on the previous solution to refine the regularization strength spatially. In this way, the model becomes progressively attentive to the image structure. For the underlying update operator, we prove the existence of a fixed point. As a special case, we investigate a mask generator for which the fixed-point iterations converge to a critical point of an explicit energy functional. In our experiments, we match the performance of state-of-the-art learned variational models for the solution of inverse problems. Additionally, we offer a promising balance between interpretability, theoretical guarantees, reliability, and performance.
{"title":"Iteratively Refined Image Reconstruction with Learned Attentive Regularizers.","authors":"Mehrsa Pourya, Sebastian Neumayer, Michael Unser","doi":"10.1080/01630563.2024.2384849","DOIUrl":"10.1080/01630563.2024.2384849","url":null,"abstract":"<p><p>We propose a regularization scheme for image reconstruction that leverages the power of deep learning while hinging on classic sparsity-promoting models. Many deep-learning-based models are hard to interpret and cumbersome to analyze theoretically. In contrast, our scheme is interpretable because it corresponds to the minimization of a series of convex problems. For each problem in the series, a mask is generated based on the previous solution to refine the regularization strength spatially. In this way, the model becomes progressively attentive to the image structure. For the underlying update operator, we prove the existence of a fixed point. As a special case, we investigate a mask generator for which the fixed-point iterations converge to a critical point of an explicit energy functional. In our experiments, we match the performance of state-of-the-art learned variational models for the solution of inverse problems. Additionally, we offer a promising balance between interpretability, theoretical guarantees, reliability, and performance.</p>","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11371266/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142134481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-11DOI: 10.1080/01630563.2024.2384869
Stefan Kindermann
We consider infinite-dimensional generalized Hilbert matrices of the form Hi,j=didjxi+xj, where di are nonnegative weights and xi are pairwise distinct positive numbers. We state sufficient and, fo...
我们考虑 Hi,j=didjxi+xj 形式的无限维广义希尔伯特矩阵，其中 di 是非负权重，xi 是成对的不同正数。我们陈述了充分的和有...
{"title":"On the Type of Ill-Posedness of Generalized Hilbert Matrices and Related Operators","authors":"Stefan Kindermann","doi":"10.1080/01630563.2024.2384869","DOIUrl":"https://doi.org/10.1080/01630563.2024.2384869","url":null,"abstract":"We consider infinite-dimensional generalized Hilbert matrices of the form Hi,j=didjxi+xj, where di are nonnegative weights and xi are pairwise distinct positive numbers. We state sufficient and, fo...","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142199962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-09DOI: 10.1080/01630563.2024.2384860
L. Olszowy, Tomasz Zając
{"title":"On some generalizations of Darbo- and Sadovskii- type fixed point theorems in Fréchet spaces","authors":"L. Olszowy, Tomasz Zając","doi":"10.1080/01630563.2024.2384860","DOIUrl":"https://doi.org/10.1080/01630563.2024.2384860","url":null,"abstract":"","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141922511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-09DOI: 10.1080/01630563.2024.2384861
Andrian Uihlein, W. Wollner
{"title":"Stabilized SQP Methods in Hilbert Spaces","authors":"Andrian Uihlein, W. Wollner","doi":"10.1080/01630563.2024.2384861","DOIUrl":"https://doi.org/10.1080/01630563.2024.2384861","url":null,"abstract":"","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141922166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-29DOI: 10.1080/01630563.2024.2349006
Yan Tang, Shiqing Zhang, Yeol Je Cho
In this paper, we focus on the solution of a class of monotone inclusion problems in reflexive Banach spaces. To reflect the geometry of the space and the operator, a more general proximal point it...
{"title":"On the Bregman-proximal iterative algorithm for the monotone inclusion problem in Banach spaces","authors":"Yan Tang, Shiqing Zhang, Yeol Je Cho","doi":"10.1080/01630563.2024.2349006","DOIUrl":"https://doi.org/10.1080/01630563.2024.2349006","url":null,"abstract":"In this paper, we focus on the solution of a class of monotone inclusion problems in reflexive Banach spaces. To reflect the geometry of the space and the operator, a more general proximal point it...","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259739","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-20DOI: 10.1080/01630563.2024.2349002
Yun-Zhang Li, Rui-Qi Dong
{"title":"Frames and Dual Frames for Krein Spaces","authors":"Yun-Zhang Li, Rui-Qi Dong","doi":"10.1080/01630563.2024.2349002","DOIUrl":"https://doi.org/10.1080/01630563.2024.2349002","url":null,"abstract":"","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141121390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-02DOI: 10.1080/01630563.2024.2333255
Yan Ni, Liu Zexian
Conjugate gradient methods are a class of very effective iterative methods for large-scale unconstrained optimization. In this paper, a new Dai-Liao conjugate gradient method for solving large-scal...
{"title":"A New Dai-Liao Conjugate Gradient Method based on Approximately Optimal Stepsize for Unconstrained Optimization","authors":"Yan Ni, Liu Zexian","doi":"10.1080/01630563.2024.2333255","DOIUrl":"https://doi.org/10.1080/01630563.2024.2333255","url":null,"abstract":"Conjugate gradient methods are a class of very effective iterative methods for large-scale unconstrained optimization. In this paper, a new Dai-Liao conjugate gradient method for solving large-scal...","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140580246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-02DOI: 10.1080/01630563.2024.2333250
Sezer Erdem, Serkan Demiriz, Adem Şahin
In the current work, it is constructed the Motzkin matrix obtained by using Motzkin numbers M=(mrs) and is examined the sequence spaces c(M) and c0(M) described as the domain of Motzkin matrix M...
在当前的工作中，利用莫兹金数 M=(mrs) 构造了莫兹金矩阵，并研究了作为莫兹金矩阵 M 域的序列空间 c(M) 和 c0(M)...
{"title":"Motzkin Sequence Spaces and Motzkin Core","authors":"Sezer Erdem, Serkan Demiriz, Adem Şahin","doi":"10.1080/01630563.2024.2333250","DOIUrl":"https://doi.org/10.1080/01630563.2024.2333250","url":null,"abstract":"In the current work, it is constructed the Motzkin matrix obtained by using Motzkin numbers M=(mrs) and is examined the sequence spaces c(M) and c0(M) described as the domain of Motzkin matrix M...","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-02DOI: 10.1080/01630563.2024.2333251
A. V. Fominyh
The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding...
本文涉及的常微分方程系统在右侧控制中包含在相位变量中不连续的控制。这些控制导致出现滑动...
{"title":"On Diferential Inclusions Arising from Some Discontinuous Systems","authors":"A. V. Fominyh","doi":"10.1080/01630563.2024.2333251","DOIUrl":"https://doi.org/10.1080/01630563.2024.2333251","url":null,"abstract":"The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding...","PeriodicalId":54707,"journal":{"name":"Numerical Functional Analysis and Optimization","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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