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We prove that quasioptimal alternating projections, when one or both projections are quasioptimal, converge locally and linearly for super-regular sets with transversal intersection. The theory is motivated by the successful application of alternating projections to low-rank matrix and tensor approximation. We focus on two problems\u2014nonnegative low-rank approximation and low-rank approximation in the maximum norm\u2014and develop fast alternating-projection algorithms for matrices and tensor trains based on cross approximation and acceleration techniques. The numerical experiments confirm that the proposed methods are efficient and suggest that they can be used to regularise various low-rank computational routines.<\/jats:p>","DOI":"10.1007\/s00211-025-01483-6","type":"journal-article","created":{"date-parts":[[2025,9,5]],"date-time":"2025-09-05T11:55:26Z","timestamp":1757073326000},"page":"1491-1535","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quasioptimal alternating projections and their use in low-rank approximation of matrices and tensors"],"prefix":"10.1007","volume":"157","author":[{"given":"Stanislav","family":"Budzinskiy","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,5]]},"reference":[{"issue":"1","key":"1483_CR1","doi-asserted-by":"publisher","first-page":"448","DOI":"10.1137\/15M1032739","volume":"26","author":"S Adly","year":"2016","unstructured":"Adly, S., Nacry, F., Thibault, L.: Preservation of prox-regularity of sets with applications to constrained optimization. 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