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This method is suitable for various statistical inference tasks, including constructing confidence intervals, inference under heteroskedastic and sub-exponential noise, and simultaneous testing. We demonstrate that the estimator achieves the Cram\u00e9r\u2013Rao lower bound on Riemannian manifolds, indicating its optimality in uncertainty quantification. We comprehensively examine statistical-to-computational gaps and investigate the impact of initialization on the minimal conditions regarding sample sizes and signal-to-noise ratios required for accurate inference. Our findings show that, with independent initialization, statistically optimal sample sizes and signal-to-noise ratios are sufficient for accurate inference. Conversely, if only dependent initialization is available, computationally optimal sample sizes and signal-to-noise ratios still guarantee asymptotic normality without the need for data-splitting. We present the phase transition between computational and statistical limits. Numerical simulation results align with the theoretical findings.<\/jats:p>","DOI":"10.1214\/25-aos2617","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:52:22Z","timestamp":1782395542000},"page":"1540-1565","source":"Crossref","is-referenced-by-count":0,"title":["Statistical inference in tensor completion: Optimal uncertainty quantification and statistical-to-computational gaps"],"prefix":"10.1214","volume":"54","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1567-0958","authenticated-orcid":false,"given":"Wanteng","family":"Ma","sequence":"first","affiliation":[{"name":"Department of Statistics and Data Science, University of Pennsylvania"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dong","family":"Xia","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hong Kong University of Science and Technology"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"40","doi-asserted-by":"publisher","unstructured":"<small>Kolda, T. 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Res.<\/i> <b>23<\/b> Paper No. [123], 77."},{"key":"2","doi-asserted-by":"publisher","unstructured":"<small>ABSIL, P.-A.<\/small> and <small>MALICK, J.<\/small> (2012). Projection-like retractions on matrix manifolds. <i>SIAM J. Optim.<\/i> <b>22<\/b> 135\u2013158.","DOI":"10.1137\/100802529"},{"key":"3","doi-asserted-by":"crossref","unstructured":"<small>AMJAD, M. J.<\/small> and <small>SHAH, D.<\/small> (2017). Censored demand estimation in retail. <i>Proc. ACM Meas. Anal. Comput. Syst.<\/i> <b>1<\/b> 1\u201328.","DOI":"10.1145\/3154489"},{"key":"4","doi-asserted-by":"crossref","unstructured":"<small>BADINGER, H.<\/small> and <small>EGGER, P.<\/small> (2013). Estimation and testing of higher-order spatial autoregressive panel data error component models. <i>J. Geogr. Syst.<\/i> <b>15<\/b> 453\u2013489.","DOI":"10.1007\/s10109-012-0174-z"},{"key":"5","unstructured":"<small>BAHADORI, M. T.<\/small>, <small>YU, Q. R.<\/small> and <small>LIU, Y.<\/small> (2014). 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Supplement to \u201cStatistical inference in tensor completion: Optimal uncertainty quantification and statistical-to-computational gaps.\u201d <a href=\"https:\/\/doi.org\/10.1214\/25-AOS2617SUPP\">https:\/\/doi.org\/10.1214\/25-AOS2617SUPP<\/a>","DOI":"10.1214\/25-AOS2617SUPP"},{"key":"50","doi-asserted-by":"crossref","unstructured":"<small>NANDPURU, H. B.<\/small>, <small>SALANKAR, S.<\/small> and <small>BORA, V.<\/small> (2014). MRI brain cancer classification using support vector machine. In 2014 <i>IEEE Students\u2019 Conference on Electrical, Electronics and Computer Science<\/i> 1\u20136. IEEE Press, New York.","DOI":"10.1109\/SCEECS.2014.6804439"},{"key":"52","unstructured":"<small>POTECHIN, A.<\/small> and <small>STEURER, D.<\/small> (2017). Exact tensor completion with sum-of-squares. In <i>Conference on Learning Theory<\/i> 1619\u20131673. 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