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

arXiv: https://arxiv.org/abs/2411.06333 · HTML: https://mlsanity.com/paper/2411.06333/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 35 references that are in the map.
- First posted on arXiv 10 Nov 2024.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Phase Retrieval using Alternating Minimization](https://mlsanity.com/paper/1306.0160/) (2013)
- [ReconFormer: Accelerated MRI Reconstruction Using Recurrent Transformer](https://mlsanity.com/paper/2201.09376/) (2022)
- [Inertial Proximal Alternating Linearized Minimization (iPALM) for Nonconvex and Nonsmooth Problems](https://mlsanity.com/paper/1702.02505/) (2017)
- [Momentum-Net: Fast and convergent iterative neural network for inverse problems](https://mlsanity.com/paper/1907.11818/) (2019)
- [Understanding Alternating Minimization for Matrix Completion](https://mlsanity.com/paper/1312.0925/) (2013)
- [A Learnable Variational Model for Joint Multimodal MRI Reconstruction and Synthesis](https://mlsanity.com/paper/2204.03804/) (2022)
- [Learnable Descent Algorithm for Nonsmooth Nonconvex Image Reconstruction](https://mlsanity.com/paper/2007.11245/) (2020)
- [Deep BCD-Net Using Identical Encoding-Decoding CNN Structures for Iterative Image Recovery](https://mlsanity.com/paper/1802.07129/) (2018)
- [Convolutional Analysis Operator Learning: Acceleration and Convergence](https://mlsanity.com/paper/1802.05584/) (2018)
- [BCD-Net for Low-dose CT Reconstruction: Acceleration, Convergence, and Generalization](https://mlsanity.com/paper/1908.01287/) (2019)
- [Provably Convergent Learned Inexact Descent Algorithm for Low-Dose CT Reconstruction](https://mlsanity.com/paper/2104.12939/) (2021)
- [Learned Alternating Minimization Algorithm for Dual-domain Sparse-View CT Reconstruction](https://mlsanity.com/paper/2306.02644/) (2023)
- [Deeply Aggregated Alternating Minimization for Image Restoration](https://mlsanity.com/paper/1612.06508/) (2016)

## Abstract (arXiv)

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.