{"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/accelerated-algorithms-for-nonlinear-matrix","title":"Accelerated Algorithms for Nonlinear Matrix Decomposition with the ReLU function","arxiv_id":"2305.08687","date":"2023-05-15","proceeding":null,"authors":["Giovanni Seraghiti","Atharva Awari","Arnaud Vandaele","Margherita Porcelli","Nicolas Gillis"],"abstract":"In this paper, we study the following nonlinear matrix decomposition (NMD) problem: given a sparse nonnegative matrix $X$, find a low-rank matrix $\\Theta$ such that $X \\approx f(\\Theta)$, where $f$ is an element-wise nonlinear function. We focus on the case where $f(\\cdot) = \\max(0, \\cdot)$, the rectified unit (ReLU) non-linear activation. We refer to the corresponding problem as ReLU-NMD. We first provide a brief overview of the existing approaches that were developed to tackle ReLU-NMD. Then we introduce two new algorithms: (1) aggressive accelerated NMD (A-NMD) which uses an adaptive Nesterov extrapolation to accelerate an existing algorithm, and (2) three-block NMD (3B-NMD) which parametrizes $\\Theta = WH$ and leads to a significant reduction in the computational cost. We also propose an effective initialization strategy based on the nuclear norm as a proxy for the rank function. We illustrate the effectiveness of the proposed algorithms (available on gitlab) on synthetic and real-world data sets.","url_abs":"https://arxiv.org/abs/2305.08687v1","url_pdf":"https://arxiv.org/pdf/2305.08687v1.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":"accelerated-algorithms-for-nonlinear-matrix","repo_url":"https://gitlab.com/ngillis/relu-nmd","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2305.08687","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}