{"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/multi-level-projection-with-exponential","title":"Multi-level projection with exponential parallel speedup; Application to sparse auto-encoders neural networks","arxiv_id":"2405.02086","date":"2024-05-03","proceeding":null,"authors":["Guillaume Perez","Michel Barlaud"],"abstract":"The $\\ell_{1,\\infty}$ norm is an efficient structured projection but the complexity of the best algorithm is unfortunately $\\mathcal{O}\\big(n m \\log(n m)\\big)$ for a matrix in $\\mathbb{R}^{n\\times m}$. In this paper, we propose a new bi-level projection method for which we show that the time complexity for the $\\ell_{1,\\infty}$ norm is only $\\mathcal{O}\\big(n m \\big)$ for a matrix in $\\mathbb{R}^{n\\times m}$, and $\\mathcal{O}\\big(n + m \\big)$ with full parallel power. We generalize our method to tensors and we propose a new multi-level projection, having an induced decomposition that yields a linear parallel speedup up to an exponential speedup factor, resulting in a time complexity lower-bounded by the sum of the dimensions, instead of the product of the dimensions. we provide a large base of implementation of our framework for bi-level and tri-level (matrices and tensors) for various norms and provides also the parallel implementation. Experiments show that our projection is $2$ times faster than the actual fastest Euclidean algorithms while providing same accuracy and better sparsity in neural networks applications.","url_abs":"https://arxiv.org/abs/2405.02086v2","url_pdf":"https://arxiv.org/pdf/2405.02086v2.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":"multi-level-projection-with-exponential","repo_url":"https://github.com/memo-p/projection","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"pytorch","reach":null}],"tasks":[],"methods":[{"method_slug":"base","method_name":"BASE"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}