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In this paper, we propose two different approaches with the aim to preserve the topological structure between the input space and the latent representation of a VAE. Firstly, we introduce InvMap-VAE as a way to turn any dimensionality reduction technique, given an embedding it produces, into a generative model within a VAE framework providing an inverse mapping into original space. Secondly, we propose the Witness Simplicial VAE as an extension of the simplicial auto-encoder to the variational setup using a witness complex for computing the simplicial regularization, and we motivate this method theoretically using tools from algebraic topology. The Witness Simplicial VAE is independent of any dimensionality reduction technique and together with its extension, Isolandmarks Witness Simplicial VAE, preserves the persistent Betti numbers of a dataset better than a standard VAE.<\/jats:p>","DOI":"10.3390\/make5010014","type":"journal-article","created":{"date-parts":[[2023,2,6]],"date-time":"2023-02-06T05:29:05Z","timestamp":1675661345000},"page":"199-236","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["InvMap and Witness Simplicial Variational Auto-Encoders"],"prefix":"10.3390","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6649-3325","authenticated-orcid":false,"given":"Aniss Aiman","family":"Medbouhi","sequence":"first","affiliation":[{"name":"Division of Robotics Perception and Learning, Department of Intelligent Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9805-0388","authenticated-orcid":false,"given":"Vladislav","family":"Polianskii","sequence":"additional","affiliation":[{"name":"Division of Robotics Perception and Learning, Department of Intelligent Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0900-1523","authenticated-orcid":false,"given":"Anastasia","family":"Varava","sequence":"additional","affiliation":[{"name":"Division of Robotics Perception and Learning, Department of Intelligent Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2965-2953","authenticated-orcid":false,"given":"Danica","family":"Kragic","sequence":"additional","affiliation":[{"name":"Division of Robotics Perception and Learning, Department of Intelligent Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,5]]},"reference":[{"key":"ref_1","unstructured":"Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N., and Weinberger, K.Q. 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