{"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/generalizing-point-embeddings-using-the","title":"Generalizing Point Embeddings using the Wasserstein Space of Elliptical Distributions","arxiv_id":"1805.07594","date":"2018-05-19","proceeding":"NeurIPS 2018 12","authors":["Boris Muzellec","Marco Cuturi"],"abstract":"Embedding complex objects as vectors in low dimensional spaces is a\nlongstanding problem in machine learning. We propose in this work an extension\nof that approach, which consists in embedding objects as elliptical probability\ndistributions, namely distributions whose densities have elliptical level sets.\nWe endow these measures with the 2-Wasserstein metric, with two important\nbenefits: (i) For such measures, the squared 2-Wasserstein metric has a closed\nform, equal to a weighted sum of the squared Euclidean distance between means\nand the squared Bures metric between covariance matrices. The latter is a\nRiemannian metric between positive semi-definite matrices, which turns out to\nbe Euclidean on a suitable factor representation of such matrices, which is\nvalid on the entire geodesic between these matrices. (ii) The 2-Wasserstein\ndistance boils down to the usual Euclidean metric when comparing Diracs, and\ntherefore provides a natural framework to extend point embeddings. We show that\nfor these reasons Wasserstein elliptical embeddings are more intuitive and\nyield tools that are better behaved numerically than the alternative choice of\nGaussian embeddings with the Kullback-Leibler divergence. In particular, and\nunlike previous work based on the KL geometry, we learn elliptical\ndistributions that are not necessarily diagonal. We demonstrate the advantages\nof elliptical embeddings by using them for visualization, to compute embeddings\nof words, and to reflect entailment or hypernymy.","url_abs":"http://arxiv.org/abs/1805.07594v5","url_pdf":"http://arxiv.org/pdf/1805.07594v5.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":"generalizing-point-embeddings-using-the","repo_url":"https://github.com/BorisMuzellec/EllipticalEmbeddings","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"generalizing-point-embeddings-using-the","repo_url":"https://github.com/albpurpura/PE4IR","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"Apache-2.0"}}],"tasks":[{"task_slug":null,"task_name":"valid"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1805.07594","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1805.07594"}},"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. 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