Recent advances in variational autoencoders (VAEs) have enabled learning latent manifolds as compact Lie groups, such as SO(d). Since this approach assumes that data lies on a subspace that is homeomorphic to the Lie group itself, we here investigate how this assumption holds in the context of images that are generated by projecting a d-dimensional volume with unknown pose in SO(d). Upon examining different theoretical candidates for the group and image space, we show that the attempt to define a group action on the data space generally fails, as it requires more specific geometric constraints on the volume. Using geometric VAEs, our experiments confirm that this constraint is key to proper pose inference, and we discuss the potential of these results for applications and future work.
{"title":"Defining an action of <i>SO</i>(<i>d</i>)-rotations on images generated by projections of <i>d</i>-dimensional objects: Applications to pose inference with Geometric VAEs.","authors":"Nicolas Legendre, Khanh Dao Duc, Nina Miolane","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>Recent advances in variational autoencoders (VAEs) have enabled learning latent manifolds as compact Lie groups, such as <i>SO</i>(<i>d</i>). Since this approach assumes that data lies on a subspace that is homeomorphic to the Lie group itself, we here investigate how this assumption holds in the context of images that are generated by projecting a <i>d</i>-dimensional volume with unknown pose in <i>SO</i>(<i>d</i>). Upon examining different theoretical candidates for the group and image space, we show that the attempt to define a group action on the data space generally fails, as it requires more specific geometric constraints on the volume. Using geometric VAEs, our experiments confirm that this constraint is key to proper pose inference, and we discuss the potential of these results for applications and future work.</p>","PeriodicalId":72637,"journal":{"name":"Colloques sur le traitement du signal et des images","volume":"28 ","pages":"329-332"},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10354539/pdf/nihms-1823247.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"10226480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-07-23DOI: 10.48550/arXiv.2207.11582
Nicolas Legendre, K. D. Duc, Nina Miolane
Recent advances in variational autoencoders (VAEs) have enabled learning latent manifolds as compact Lie groups, such as SO(d). Since this approach assumes that data lies on a subspace that is homeomorphic to the Lie group itself, we here investigate how this assumption holds in the context of images that are generated by projecting a d-dimensional volume with unknown pose in SO(d). Upon examining different theoretical candidates for the group and image space, we show that the attempt to define a group action on the data space generally fails, as it requires more specific geometric constraints on the volume. Using geometric VAEs, our experiments confirm that this constraint is key to proper pose inference, and we discuss the potential of these results for applications and future work.
{"title":"Defining an action of SO(d)-rotations on images generated by projections of d-dimensional objects: Applications to pose inference with Geometric VAEs","authors":"Nicolas Legendre, K. D. Duc, Nina Miolane","doi":"10.48550/arXiv.2207.11582","DOIUrl":"https://doi.org/10.48550/arXiv.2207.11582","url":null,"abstract":"Recent advances in variational autoencoders (VAEs) have enabled learning latent manifolds as compact Lie groups, such as SO(d). Since this approach assumes that data lies on a subspace that is homeomorphic to the Lie group itself, we here investigate how this assumption holds in the context of images that are generated by projecting a d-dimensional volume with unknown pose in SO(d). Upon examining different theoretical candidates for the group and image space, we show that the attempt to define a group action on the data space generally fails, as it requires more specific geometric constraints on the volume. Using geometric VAEs, our experiments confirm that this constraint is key to proper pose inference, and we discuss the potential of these results for applications and future work.","PeriodicalId":72637,"journal":{"name":"Colloques sur le traitement du signal et des images","volume":"5 1","pages":"329-332"},"PeriodicalIF":0.0,"publicationDate":"2022-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79677521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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