DEs-Inspired Accelerated Unfolded Linearized ADMM Networks for Inverse Problems

Area of Science:

• Optimization Algorithms
• Deep Learning
• Applied Mathematics

Background:

• Traditional alternating direction multiplier methods (ADMMs) are increasingly understood through continuous-time differential equations (DEs).
• Unfolded deep networks inherit ADMM iterations but lack clear structural insights.
• Existing unfolded methods show practical performance gains but limited theoretical understanding.

Purpose of the Study:

• To explore unfolded linearized ADMM (LADMM) from a differential equation (DE) perspective.
• To design novel, more efficient unfolded deep networks based on DE insights.
• To establish theoretical guarantees connecting unfolded ADMMs with DEs.

Main Methods:

• Proposed an unfolded Euler LADMM scheme and a more accurate Trapezoid LADMM scheme based on trapezoid discretization.
• Developed explicit versions of the Trapezoid LADMM scheme using a prediction-correction strategy.
• Designed accelerated Euler and Trapezoid LADMM variants, interpretable as second-order DEs, to expand network representation capabilities.
• Implemented schemes as (A-)ELADMM and (A-)TLADMM with proximal operators and (A-)ELADMM-Net and (A-)TLADMM-Net with convolutional neural networks (CNNs).

Main Results:

• Demonstrated a comprehensive connection between unfolded ADMMs and first- (second-) order DEs with theoretical guarantees.
• The proposed Trapezoid LADMM schemes (A-TLADMM) showed superior performance in extensive inverse problem experiments compared to existing methods.
• Accelerated schemes expanded the representation space of unfolded networks, enhancing capability.

Conclusions:

• The study provides the first theoretical framework linking unfolded ADMMs to DEs, offering insights into network structures.
• The novel Trapezoid LADMM schemes and their accelerated variants significantly improve performance in inverse problems.
• This work paves the way for designing more efficient and interpretable deep learning models for optimization tasks.