{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T18:40:51Z","timestamp":1776364851195,"version":"3.51.2"},"reference-count":33,"publisher":"Wiley","issue":"6","license":[{"start":{"date-parts":[[2023,6,5]],"date-time":"2023-06-05T00:00:00Z","timestamp":1685923200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008982","name":"National Science Foundation of Sri Lanka","doi-asserted-by":"publisher","award":["CCF\u20101942892"],"award-info":[{"award-number":["CCF\u20101942892"]}],"id":[{"id":"10.13039\/501100008982","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pericles.pericles-gcp.literatumonline.com"],"crossmark-restriction":true},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low\u2010rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill\u2010conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP\u2010ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP\u2010ALS subproblems efficiently, have the same complexity as the standard CP\u2010ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill\u2010conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.<\/jats:p>","DOI":"10.1002\/nla.2511","type":"journal-article","created":{"date-parts":[[2023,6,7]],"date-time":"2023-06-07T18:40:55Z","timestamp":1686163255000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["CP decomposition for tensors via alternating least squares with QR decomposition"],"prefix":"10.1002","volume":"30","author":[{"given":"Rachel","family":"Minster","sequence":"first","affiliation":[{"name":"Department of Computer Science Wake Forest University  Winston\u2010Salem North Carolina USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Irina","family":"Viviano","sequence":"additional","affiliation":[{"name":"Clinical &amp; Translational Science Institute Wake Forest University School of Medicine  Winston\u2010Salem North Carolina USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9369-4029","authenticated-orcid":false,"given":"Xiaotian","family":"Liu","sequence":"additional","affiliation":[{"name":"Department of Computer Science Wake Forest University  Winston\u2010Salem North Carolina USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Grey","family":"Ballard","sequence":"additional","affiliation":[{"name":"Department of Computer Science Wake Forest University  Winston\u2010Salem North Carolina USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2023,6,5]]},"reference":[{"key":"e_1_2_8_2_1","unstructured":"BaderBW KoldaTG et al.Tensor Toolbox for MATLAB Version 3.2.1. 2021. 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