{"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/explorations-in-homeomorphic-variational-auto","title":"Explorations in Homeomorphic Variational Auto-Encoding","arxiv_id":"1807.04689","date":"2018-07-12","proceeding":null,"authors":["Luca Falorsi","Pim de Haan","Tim R. Davidson","Nicola De Cao","Maurice Weiler","Patrick Forré","Taco S. Cohen"],"abstract":"The manifold hypothesis states that many kinds of high-dimensional data are\nconcentrated near a low-dimensional manifold. If the topology of this data\nmanifold is non-trivial, a continuous encoder network cannot embed it in a\none-to-one manner without creating holes of low density in the latent space.\nThis is at odds with the Gaussian prior assumption typically made in\nVariational Auto-Encoders (VAEs), because the density of a Gaussian\nconcentrates near a blob-like manifold.\n  In this paper we investigate the use of manifold-valued latent variables.\nSpecifically, we focus on the important case of continuously differentiable\nsymmetry groups (Lie groups), such as the group of 3D rotations\n$\\operatorname{SO}(3)$. We show how a VAE with $\\operatorname{SO}(3)$-valued\nlatent variables can be constructed, by extending the reparameterization trick\nto compact connected Lie groups. Our experiments show that choosing\nmanifold-valued latent variables that match the topology of the latent data\nmanifold, is crucial to preserve the topological structure and learn a\nwell-behaved latent space.","url_abs":"http://arxiv.org/abs/1807.04689v1","url_pdf":"http://arxiv.org/pdf/1807.04689v1.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":"explorations-in-homeomorphic-variational-auto","repo_url":"https://github.com/pimdh/lie-vae","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1807.04689","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1807.04689"}},"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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