{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:42:36Z","timestamp":1759970556366,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,28]],"date-time":"2025-01-28T00:00:00Z","timestamp":1738022400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Sichuan Province","award":["2023NSFSC0020","2024NSFSC0083"],"award-info":[{"award-number":["2023NSFSC0020","2024NSFSC0083"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we extend the definition of tensor products from using an invertible matrix to utilising right-invertible matrices, exploring the algebraic properties of these new tensor products. Based on this novel definition, we define the concepts of tensor rank and tensor nuclear norm, ensuring consistency with their matrix counterparts, and derive a singular value thresholding (\u2217L,R SVT) formula to approximately solve the subproblems in the alternating direction method of multipliers (ADMM), which is integral to our proposed tensor robust principal component analysis (\u2217LR TRPCA) algorithm. The computational complexity of the \u2217LR TRPCA algorithm is O(k\u00b7(n1n2n3+p\u00b7min(n12n2,n1n22))) for k iterations. According to this complexity analysis, by using a right-invertible matrix that selects p rows from the n3 rows of the invertible matrix used in the tensor product with an invertible matrix, the computational load is approximately reduced to p\/n3 of what it would be with an invertible matrix, highlighting the efficiency gain in terms of computational resources. We apply this efficient algorithm to grayscale video denoising and motion detection problems, where it demonstrates significant improvements in processing speed while maintaining comparable quality levels to existing methods, thereby providing a promising approach for handling multi-linear data and offering valuable insights for advanced data analysis tasks.<\/jats:p>","DOI":"10.3390\/axioms14020099","type":"journal-article","created":{"date-parts":[[2025,1,28]],"date-time":"2025-01-28T08:57:48Z","timestamp":1738054668000},"page":"99","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient Tensor Robust Principal Analysis via Right-Invertible Matrix-Based Tensor Products"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-1687-5461","authenticated-orcid":false,"given":"Zhang","family":"Huang","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8066-5261","authenticated-orcid":false,"given":"Jun","family":"Feng","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,28]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Robust principal component analysis?","volume":"58","author":"Li","year":"2011","journal-title":"J. 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