A Learned Proximal Alternating Minimization Algorithm and Its Induced Network for a Class of Two-Block Nonconvex and Nonsmooth Optimization

This work proposes a general learned proximal alternating minimization algorithm, LPAM, for solving learnable two-block nonsmooth and nonconvex optimization problems. We tackle the nonsmoothness by an appropriate smoothing technique with automatic diminishing smoothing effect. For smoothed nonconvex...

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Vydané v:	Journal of scientific computing Ročník 103; číslo 2; s. 56
Hlavní autori:	Chen, Yunmei, Liu, Lezhi, Zhang, Lei
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.05.2025 Springer Nature B.V
Predmet:	Accumulation Algorithms Computational Mathematics and Numerical Analysis Computer vision Convergence Inverse problems Learning Machine learning Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Neural networks Optimization Optimization algorithms Smoothing Sparsity Theoretical Deep learning 68U10 65K05 MRI Image reconstruction Learned alternating minimization algorithm Nonconvex nonsmooth optimization 65K10 90C26
ISSN:	0885-7474, 1573-7691
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Abstract	This work proposes a general learned proximal alternating minimization algorithm, LPAM, for solving learnable two-block nonsmooth and nonconvex optimization problems. We tackle the nonsmoothness by an appropriate smoothing technique with automatic diminishing smoothing effect. For smoothed nonconvex problems we modify the proximal alternating linearized minimization (PALM) scheme by incorporating the residual learning architecture, which has proven to be highly effective in deep network training, and employing the block coordinate decent (BCD) iterates as a safeguard for the convergence of the algorithm. We prove that there is a subsequence of the iterates generated by LPAM, which has at least one accumulation point and each accumulation point is a Clarke stationary point. Our method is widely applicable as one can employ various learning problems formulated as two-block optimizations, and is also easy to be extended for solving multi-block nonsmooth and nonconvex optimization problems. The network, whose architecture follows the LPAM exactly, namely LPAM-net, inherits the convergence properties of the algorithm to make the network interpretable. As an example application of LPAM-net, we present the numerical and theoretical results on the application of LPAM-net for joint multi-modal MRI reconstruction with significantly under-sampled k -space data. The experimental results indicate the proposed LPAM-net is parameter-efficient and has favourable performance in comparison with some state-of-the-art methods.
AbstractList	This work proposes a general learned proximal alternating minimization algorithm, LPAM, for solving learnable two-block nonsmooth and nonconvex optimization problems. We tackle the nonsmoothness by an appropriate smoothing technique with automatic diminishing smoothing effect. For smoothed nonconvex problems we modify the proximal alternating linearized minimization (PALM) scheme by incorporating the residual learning architecture, which has proven to be highly effective in deep network training, and employing the block coordinate decent (BCD) iterates as a safeguard for the convergence of the algorithm. We prove that there is a subsequence of the iterates generated by LPAM, which has at least one accumulation point and each accumulation point is a Clarke stationary point. Our method is widely applicable as one can employ various learning problems formulated as two-block optimizations, and is also easy to be extended for solving multi-block nonsmooth and nonconvex optimization problems. The network, whose architecture follows the LPAM exactly, namely LPAM-net, inherits the convergence properties of the algorithm to make the network interpretable. As an example application of LPAM-net, we present the numerical and theoretical results on the application of LPAM-net for joint multi-modal MRI reconstruction with significantly under-sampled k -space data. The experimental results indicate the proposed LPAM-net is parameter-efficient and has favourable performance in comparison with some state-of-the-art methods. This work proposes a general learned proximal alternating minimization algorithm, LPAM, for solving learnable two-block nonsmooth and nonconvex optimization problems. We tackle the nonsmoothness by an appropriate smoothing technique with automatic diminishing smoothing effect. For smoothed nonconvex problems we modify the proximal alternating linearized minimization (PALM) scheme by incorporating the residual learning architecture, which has proven to be highly effective in deep network training, and employing the block coordinate decent (BCD) iterates as a safeguard for the convergence of the algorithm. We prove that there is a subsequence of the iterates generated by LPAM, which has at least one accumulation point and each accumulation point is a Clarke stationary point. Our method is widely applicable as one can employ various learning problems formulated as two-block optimizations, and is also easy to be extended for solving multi-block nonsmooth and nonconvex optimization problems. The network, whose architecture follows the LPAM exactly, namely LPAM-net, inherits the convergence properties of the algorithm to make the network interpretable. As an example application of LPAM-net, we present the numerical and theoretical results on the application of LPAM-net for joint multi-modal MRI reconstruction with significantly under-sampled k-space data. The experimental results indicate the proposed LPAM-net is parameter-efficient and has favourable performance in comparison with some state-of-the-art methods.
ArticleNumber	56
Author	Zhang, Lei Liu, Lezhi Chen, Yunmei
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Cites_doi	10.1007/978-3-030-32226-7_4 10.1007/978-3-031-16446-0_34 10.1137/120887679 10.1109/IVMSPW.2018.8448694 10.1109/TIP.2019.2937734 10.1145/2488608.2488693 10.1109/FOCS.2014.75 10.1109/TMI.2023.3314747 10.48550/ARXIV.2104.12939 10.1109/TMI.2018.2865356 10.1287/moor.1100.0449 10.1016/0041-5553(64)90137-5 10.1109/TMI.2023.3314008 10.1007/s10107-011-0484-9 10.1007/s10107-013-0701-9 10.1007/978-3-031-43999-5_17 10.1109/CVPR.2016.90 10.1109/CVPR.2017.38 10.1109/ICASSP.2009.4960116 10.1137/20M1353368 10.1137/16M1064064 10.1016/j.jmr.2022.107354 10.1109/TIP.2011.2175740 10.1002/mp.14006 10.1016/j.sigpro.2011.10.012 10.1109/83.551699 10.1109/TMI.2014.2377694 10.1007/978-3-319-46493-0_38 10.1007/978-3-319-91578-4
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SubjectTerms	Accumulation Algorithms Computational Mathematics and Numerical Analysis Computer vision Convergence Inverse problems Learning Machine learning Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Neural networks Optimization Optimization algorithms Smoothing Sparsity Theoretical
Title	A Learned Proximal Alternating Minimization Algorithm and Its Induced Network for a Class of Two-Block Nonconvex and Nonsmooth Optimization
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