{"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/an-extrapolated-and-provably-convergent","title":"An extrapolated and provably convergent algorithm for nonlinear matrix decomposition with the ReLU function","arxiv_id":"2503.23832","date":"2025-03-31","proceeding":null,"authors":["Nicolas Gillis","Margherita Porcelli","Giovanni Seraghiti"],"abstract":"Nonlinear matrix decomposition (NMD) with the ReLU function, denoted ReLU-NMD, is the following problem: given a sparse, nonnegative matrix $X$ and a factorization rank $r$, identify a rank-$r$ matrix $\\Theta$ such that $X\\approx \\max(0,\\Theta)$. This decomposition finds application in data compression, matrix completion with entries missing not at random, and manifold learning. The standard ReLU-NMD model minimizes the least squares error, that is, $\\|X - \\max(0,\\Theta)\\|_F^2$. The corresponding optimization problem is nondifferentiable and highly nonconvex. This motivated Saul to propose an alternative model, Latent-ReLU-NMD, where a latent variable $Z$ is introduced and satisfies $\\max(0,Z)=X$ while minimizing $\\|Z - \\Theta\\|_F^2$ (``A nonlinear matrix decomposition for mining the zeros of sparse data'', SIAM J. Math. Data Sci., 2022). Our first contribution is to show that the two formulations may yield different low-rank solutions $\\Theta$; in particular, we show that Latent-ReLU-NMD can be ill-posed when ReLU-NMD is not, meaning that there are instances in which the infimum of Latent-ReLU-NMD is not attained while that of ReLU-NMD is. We also consider another alternative model, called 3B-ReLU-NMD, which parameterizes $\\Theta=WH$, where $W$ has $r$ columns and $H$ has $r$ rows, allowing one to get rid of the rank constraint in Latent-ReLU-NMD. Our second contribution is to prove the convergence of a block coordinate descent (BCD) applied to 3B-ReLU-NMD and referred to as BCD-NMD. Our third contribution is a novel extrapolated variant of BCD-NMD, dubbed eBCD-NMD, which we prove is also convergent under mild assumptions. We illustrate the significant acceleration effect of eBCD-NMD compared to BCD-NMD, and also show that eBCD-NMD performs well against the state of the art on synthetic and real-world data sets.","url_abs":"https://arxiv.org/abs/2503.23832v1","url_pdf":"https://arxiv.org/pdf/2503.23832v1.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":"an-extrapolated-and-provably-convergent","repo_url":"https://github.com/giovanniseraghiti/relu-nmd","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"data-compression","task_name":"Data Compression"},{"task_slug":"math","task_name":"Math"},{"task_slug":"matrix-completion","task_name":"Matrix Completion"}],"methods":[{"method_slug":"relu","method_name":"ReLU"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}