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The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling intractable random variables with tractable reference random variables. This paper generalises the functional tensor-train approximation of the inverse Rosenblatt transport recently developed by Dolgov et al. (Stat Comput 30:603\u2013625, 2020) to a wide class of high-dimensional non-negative functions, such as unnormalised probability density functions. First, we extend the inverse Rosenblatt transform to enable the transport to general reference measures other than the uniform measure. We develop an efficient procedure to compute this transport from a squared tensor-train decomposition which preserves the monotonicity. More crucially, we integrate the proposed order-preserving functional tensor-train transport into a nested variable transformation framework inspired by the layered structure of deep neural networks. The resulting deep inverse Rosenblatt transport significantly expands the capability of tensor approximations and transport maps to random variables with complicated nonlinear interactions and concentrated density functions. We demonstrate the efficiency of the proposed approach on a range of applications in statistical learning and uncertainty quantification, including parameter estimation for dynamical systems and inverse problems constrained by partial differential equations.<\/jats:p>","DOI":"10.1007\/s10208-021-09537-5","type":"journal-article","created":{"date-parts":[[2021,9,21]],"date-time":"2021-09-21T16:44:09Z","timestamp":1632242649000},"page":"1863-1922","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports"],"prefix":"10.1007","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4840-8545","authenticated-orcid":false,"given":"Tiangang","family":"Cui","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1647-4214","authenticated-orcid":false,"given":"Sergey","family":"Dolgov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,9,21]]},"reference":[{"key":"9537_CR1","doi-asserted-by":"crossref","unstructured":"Maxime Barrault, Yvon Maday, Ngoc\u00a0Cuong Nguyen, and Anthony\u00a0T Patera, An \u2018empirical interpolation\u2019ethod: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique 339 (2004), no.\u00a09, 667\u2013672.","DOI":"10.1016\/j.crma.2004.08.006"},{"issue":"4","key":"9537_CR2","doi-asserted-by":"publisher","first-page":"A2405","DOI":"10.1137\/15M1036919","volume":"38","author":"Daniele 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