{"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/differentiating-through-the-frechet-mean","title":"Differentiating through the Fréchet Mean","arxiv_id":"2003.00335","date":"2020-02-29","proceeding":"ICML 2020 1","authors":["Aaron Lou","Isay Katsman","Qingxuan Jiang","Serge Belongie","Ser-Nam Lim","Christopher De Sa"],"abstract":"Recent advances in deep representation learning on Riemannian manifolds extend classical deep learning operations to better capture the geometry of the manifold. One possible extension is the Fr\\'echet mean, the generalization of the Euclidean mean; however, it has been difficult to apply because it lacks a closed form with an easily computable derivative. In this paper, we show how to differentiate through the Fr\\'echet mean for arbitrary Riemannian manifolds. Then, focusing on hyperbolic space, we derive explicit gradient expressions and a fast, accurate, and hyperparameter-free Fr\\'echet mean solver. This fully integrates the Fr\\'echet mean into the hyperbolic neural network pipeline. To demonstrate this integration, we present two case studies. First, we apply our Fr\\'echet mean to the existing Hyperbolic Graph Convolutional Network, replacing its projected aggregation to obtain state-of-the-art results on datasets with high hyperbolicity. Second, to demonstrate the Fr\\'echet mean's capacity to generalize Euclidean neural network operations, we develop a hyperbolic batch normalization method that gives an improvement parallel to the one observed in the Euclidean setting.","url_abs":"https://arxiv.org/abs/2003.00335v4","url_pdf":"https://arxiv.org/pdf/2003.00335v4.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":"differentiating-through-the-frechet-mean","repo_url":"https://github.com/CUAI/Differentiable-Frechet-Mean","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"differentiating-through-the-frechet-mean","repo_url":"https://github.com/CUVL/Differentiable-Frechet-Mean","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"representation-learning","task_name":"Representation Learning"}],"methods":[{"method_slug":"batch-normalization","method_name":"Batch Normalization"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2003.00335","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2003.00335"}},"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/CUAI/Differentiable-Frechet-Mean","reach":{"status":"ok","spdx":"MIT"}},{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/CUVL/Differentiable-Frechet-Mean","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"ran_draft_wrong":1,"unverified":2},"by_repo_kind":{"official":{"samples":2,"ran":0,"repositories":1},"listed":{"samples":1,"ran":1,"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":0,"samples":[{"code_sha256_prefix":"189316d1476b8b6b","entry":"tanh","repo":"CUVL/Differentiable-Frechet-Mean","repo_kind":"listed","path":"frechetmean/utils.py","file_url":"https://github.com/CUVL/Differentiable-Frechet-Mean/blob/HEAD/frechetmean/utils.py","link_basis":"plan_row","language":"python","status":"ran_draft_wrong","verification_level":1,"contract_check":"MISDECLARED","metamorphic_tier":null,"behaviour_fingerprint":true,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"189316d1476b8b6b"}},{"code_sha256_prefix":"770cebfcb5d6aaa0","entry":"cosh","repo":"CUAI/Differentiable-Frechet-Mean","repo_kind":"official","path":"frechetmean/utils.py","file_url":"https://github.com/CUAI/Differentiable-Frechet-Mean/blob/HEAD/frechetmean/utils.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"770cebfcb5d6aaa0"}},{"code_sha256_prefix":"e02a905fcb5a071d","entry":"sinh","repo":"CUAI/Differentiable-Frechet-Mean","repo_kind":"official","path":"frechetmean/utils.py","file_url":"https://github.com/CUAI/Differentiable-Frechet-Mean/blob/HEAD/frechetmean/utils.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"e02a905fcb5a071d"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}