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Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>The canonical tensor rank approximation problem (TAP) consists of approximating a real-valued tensor by one of low canonical rank, which is a challenging nonlinear, nonconvex, constrained optimization problem, where the constraint set forms a nonsmooth semialgebraic set. We introduce a Riemannian Gauss--Newton method with trust region for solving small-scale, dense TAPs. The novelty of our approach is threefold. First, we parametrize the constraint set as the Cartesian product of Segre manifolds, hereby formulating the TAP as a Riemannian optimization problem, and we argue why this parametrization is theoretically a good choice. Second, an original sequentially truncated higher-order singular value decomposition (ST-HOSVD) based retraction operator is proposed. Third, we introduce a hot restart mechanism that efficiently detects when the optimization process is tending to an ill-conditioned tensor rank decomposition and that often yields a quick escape path from such spurious decompositions. Numerical experiments show improvements of up to three orders of magnitude in terms of the expected time to compute a successful solution over existing state-of-the-art methods.<\/jats:p>","DOI":"10.1137\/17m114618x","type":"journal-article","created":{"date-parts":[[2018,9,11]],"date-time":"2018-09-11T16:35:27Z","timestamp":1536683727000},"page":"2435-2465","source":"Crossref","is-referenced-by-count":38,"title":["A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem"],"prefix":"10.1137","volume":"28","author":[{"given":"Paul","family":"Breiding","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5692-4163","authenticated-orcid":true,"given":"Nick","family":"Vannieuwenhoven","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,9,11]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"P.A. 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