{"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/simplifying-momentum-based-riemannian","title":"Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning","arxiv_id":"2302.09738","date":"2023-02-20","proceeding":null,"authors":["Wu Lin","Valentin Duruisseaux","Melvin Leok","Frank Nielsen","Mohammad Emtiyaz Khan","Mark Schmidt"],"abstract":"Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of sparse or structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free $2^\\text{nd}$-order optimizers for deep learning with low precision by using only matrix multiplications. Code: https://github.com/yorkerlin/StructuredNGD-DL","url_abs":"https://arxiv.org/abs/2302.09738v9","url_pdf":"https://arxiv.org/pdf/2302.09738v9.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":"simplifying-momentum-based-riemannian","repo_url":"https://github.com/yorkerlin/structuredngd-dl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2302.09738","atlas_url":"https://app.syntology.ai/?focus=2302.09738","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}