Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers
This work addresses image reconstruction problems in domains like biomedical imaging, offering improved performance over existing methods, though it is incremental as it builds on known concepts like polyhedral norms and convex regularization.
The paper tackles learning convex regularizers for image reconstruction from limited data by narrowing admissible functionals to powers of seminorms and approximating them with polyhedral norms, showing that the proposed framework outperforms sparsity-based compressed sensing methods with similar convergence and robustness guarantees.
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.