{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/l_p-norm-constrained-coding-with-frank-wolfe","title":"Frank-Wolfe Network: An Interpretable Deep Structure for Non-Sparse Coding","arxiv_id":"1802.10252","date":"2018-02-28","proceeding":null,"authors":["Dong Liu","Ke Sun","Zhangyang Wang","Runsheng Liu","Zheng-Jun Zha"],"abstract":"The problem of $L_p$-norm constrained coding is to convert signal into code that lies inside an $L_p$-ball and most faithfully reconstructs the signal. Previous works under the name of sparse coding considered the cases of $L_0$ and $L_1$ norms. The cases with $p>1$ values, i.e. non-sparse coding studied in this paper, remain a difficulty. We propose an interpretable deep structure namely Frank-Wolfe Network (F-W Net), whose architecture is inspired by unrolling and truncating the Frank-Wolfe algorithm for solving an $L_p$-norm constrained problem with $p\\geq 1$. We show that the Frank-Wolfe solver for the $L_p$-norm constraint leads to a novel closed-form nonlinear unit, which is parameterized by $p$ and termed $pool_p$. The $pool_p$ unit links the conventional pooling, activation, and normalization operations, making F-W Net distinct from existing deep networks either heuristically designed or converted from projected gradient descent algorithms. We further show that the hyper-parameter $p$ can be made learnable instead of pre-chosen in F-W Net, which gracefully solves the non-sparse coding problem even with unknown $p$. We evaluate the performance of F-W Net on an extensive range of simulations as well as the task of handwritten digit recognition, where F-W Net exhibits strong learning capability. We then propose a convolutional version of F-W Net, and apply the convolutional F-W Net into image denoising and super-resolution tasks, where F-W Net all demonstrates impressive effectiveness, flexibility, and robustness.","url_abs":"https://arxiv.org/abs/1802.10252v4","url_pdf":"https://arxiv.org/pdf/1802.10252v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"l_p-norm-constrained-coding-with-frank-wolfe","repo_url":"https://github.com/sunke123/FW-Net","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"denoising","task_name":"Denoising"},{"task_slug":"handwritten-digit-recognition","task_name":"Handwritten Digit Recognition"},{"task_slug":"image-denoising","task_name":"Image Denoising"},{"task_slug":"unrolling","task_name":"Rolling Shutter Correction"},{"task_slug":"super-resolution","task_name":"Super-Resolution"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}