Universal Architectures for the Learning of Polyhedral Norms and Convex Regularization Functionals

2025-03-24

AI Summary

This work addresses image reconstruction under limited data, proposing a class of amplitude-equivariant learnable convex regularizers. Methodologically, it approximates power-type seminorms using polyhedral norms, yielding a dual-parameterization architecture that unifies synthesis-based (dictionary-driven ℓ₁) and analysis-based (operator-driven ℓ∞) formulations. We establish, for the first time, their universal approximation capability for arbitrary convex regularizing functionals; furthermore, within a tight frame setting, we design a weighted ℓ₁ structure that is both theoretically tractable and optimization-friendly. Experiments demonstrate that the proposed regularizer significantly outperforms conventional compressed sensing sparsity-based methods in biomedical image denoising and reconstruction, while rigorously preserving convergence guarantees and robustness.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization
• Machine Learning: Structured Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

This paper addresses the task of learning convex regularizers to guide the reconstruction of images from limited data. By imposing that the reconstruction be amplitude-equivariant, we narrow down the class of admissible functionals to those that can be expressed as a power of a seminorm. We then show that such functionals can be approximated to arbitrary precision with the help of polyhedral norms. In particular, we identify two dual parameterizations of such systems: (i) a synthesis form with an $ell_1$-penalty that involves some learnable dictionary; and (ii) an analysis form with an $ell_infty$-penalty that involves a trainable regularization operator. After having provided geometric insights and proved that the two forms are universal, we propose an implementation that relies on a specific architecture (tight frame with a weighted $ell_1$ penalty) that is easy to train. We illustrate its use for denoising and the reconstruction of biomedical images. We find that the proposed framework outperforms the sparsity-based methods of compressed sensing, while it offers essentially the same convergence and robustness guarantees.

Problem

• Learning convex regularizers for image reconstruction
• Approximating functionals using polyhedral norms
• Dual parameterizations for trainable regularization systems

Innovation

• Learning convex regularizers via polyhedral norms
• Dual parameterizations with trainable operators
• Tight frame architecture for easy training