What's in a Prior? Learned Proximal Networks for Inverse Problems

Area of Science:

• Computational imaging
• Machine learning for inverse problems
• Optimization theory

Background:

• Proximal operators are crucial for regularizing ill-posed inverse problems.
• Deep learning models (plug-and-play, deep unrolling) approximate proximal operators but lack theoretical guarantees.
• Current data-driven methods hinder convergence analysis and understanding of learned priors.

Purpose of the Study:

• Introduce a framework for learned proximal networks (LPNs).
• Prove LPNs yield exact proximal operators for data-driven regularizers.
• Develop a training strategy (proximal matching) to recover data distribution priors.

Main Methods:

• Developed a framework for learned proximal networks (LPNs).
• Proved theoretical guarantees for LPNs as exact proximal operators.
• Introduced and analyzed the proximal matching training strategy.

Main Results:

• Learned proximal networks (LPNs) provide exact proximal operators for nonconvex regularizers.
• Proximal matching training provably recovers the log-prior of the data distribution.
• LPNs offer general, unsupervised, and expressive proximal operators for inverse problems.

Conclusions:

• LPNs provide a principled deep learning approach to proximal operators in inverse problems.
• The proximal matching strategy enables convergence guarantees and interpretable prior learning.
• Demonstrated state-of-the-art performance and insights into learned priors from data.