{"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/learning-to-normalize-on-the-spd-manifold-1","title":"Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry","arxiv_id":"2504.00660","date":"2025-04-01","proceeding":"CVPR 2025 1","authors":["Rui Wang","Shaocheng Jin","Ziheng Chen","Xiaoqing Luo","Xiao-Jun Wu"],"abstract":"Covariance matrices have proven highly effective across many scientific fields. Since these matrices lie within the Symmetric Positive Definite (SPD) manifold - a Riemannian space with intrinsic non-Euclidean geometry, the primary challenge in representation learning is to respect this underlying geometric structure. Drawing inspiration from the success of Euclidean deep learning, researchers have developed neural networks on the SPD manifolds for more faithful covariance embedding learning. A notable advancement in this area is the implementation of Riemannian batch normalization (RBN), which has been shown to improve the performance of SPD network models. Nonetheless, the Riemannian metric beneath the existing RBN might fail to effectively deal with the ill-conditioned SPD matrices (ICSM), undermining the effectiveness of RBN. In contrast, the Bures-Wasserstein metric (BWM) demonstrates superior performance for ill-conditioning. In addition, the recently introduced Generalized BWM (GBWM) parameterizes the vanilla BWM via an SPD matrix, allowing for a more nuanced representation of vibrant geometries of the SPD manifold. Therefore, we propose a novel RBN algorithm based on the GBW geometry, incorporating a learnable metric parameter. Moreover, the deformation of GBWM by matrix power is also introduced to further enhance the representational capacity of GBWM-based RBN. Experimental results on different datasets validate the effectiveness of our proposed method.","url_abs":"https://arxiv.org/abs/2504.00660v1","url_pdf":"https://arxiv.org/pdf/2504.00660v1.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":"learning-to-normalize-on-the-spd-manifold-1","repo_url":"https://github.com/jjscc/gbwbn","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"representation-learning","task_name":"Representation Learning"}],"methods":[{"method_slug":"batch-normalization","method_name":"Batch Normalization"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2504.00660","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2504.00660"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"deterministic:regex_extraction","url":"https://github.com/jjscc/GBWBN","reach":null}],"summary":{"unverified":1},"by_repo_kind":{"listed":{"samples":1,"ran":0,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":1,"samples":[{"code_sha256_prefix":"951cf61c2e9c213d","entry":"BatchNormSPD","repo":"jjscc/GBWBN","repo_kind":"listed","path":"GBWBN_SPDNet/spd/nn.py","file_url":"https://github.com/jjscc/GBWBN/blob/HEAD/GBWBN_SPDNet/spd/nn.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"951cf61c2e9c213d"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}