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

TL;DR

It is proved 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, and the proposed LPAM-net is parameter-efficient and has favourable performance in comparison with some state-of-the-art methods.

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.

Figures (12)

• Figure 1: The architecture of the propsoed Initialization network and the LPAM-net. Top: The proposed Initialization network; Middle: The architecture of the LPAM-net for joint reconstruction of T1 and T2 images; Bottom: The detailed illustration of the $k^{th}$--phase of the LPAM-net.
• Figure 2: From Left to Right: The PSNR values for the reconstruction results of the T1 and T2 image with a $20\%$ under-sampling ratio at various phases. In both figures, the LPAM-net learns the network parameters in the first 15 phases. Starting from $\mathbf{X}_{16}$ the reconstruction results are obtained from the same model \ref{['modell']} with fixed $\mathbf{g}$ learned from the 15-phase LPAM-net. The PSNR values for the reconstruction results of the first 15 phases are plotted in blue, and since the $16^{th}$ phase, that are plotted in orange.
• Figure 3: Objective function value ($\Phi(\mathbf{x}_1^{k}, \mathbf{x}_2^{k})$) of images reconstruction with a 20% under-sampling ratio across various phase number K. The results of the first 15 phases are plotted in blue and since the $16^{th}$ phase, that are plotted in orange.
• Figure 4: Top row: Representation brain T1 MRI images reconstructed by the LPAM-net with a 20% under-sampling ratio in the radial mask after 15, 150, 1005 and 5010 iterations; Bottom row: Representation brain T2 MRI images reconstructed by the LPAM-net with a 20% under-sampling ratio in radial mask after 15, 150, 1005 and 5010 iterations. PSNRs (dB) are shown in the parentheses.
• Figure 5: PSNR for the reconstruction result of images with a 10% under-sampling ratio across various phase number $K$. Left: T1 images. Right: T2 images. In both figures, the blue line represents results for joint reconstruction via LPAM-net and the orange line represents the results from the Individual-modality Reconstruction Network. The vertical line represents the standard deviation of PSNR.

Theorems & Definitions (7)

• remark 1
• definition 1
• definition 2
• lemma 1
• theorem 1
• remark 2
• remark 3