{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:52:27Z","timestamp":1791323547182,"version":"4.3.1"},"reference-count":50,"publisher":"Institute of Mathematical Statistics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:p>In this paper, we consider inference and uncertainty quantification for low Tucker rank tensors with additive noise in the high-dimensional regime. Focusing on the output of the higher-order orthogonal iteration (HOOI) algorithm, a commonly used algorithm for tensor singular value decomposition, we establish nonasymptotic distributional theory and study how to construct confidence regions and intervals for both the estimated singular vectors and the tensor entries in the presence of heteroskedastic subgaussian noise, which are further shown to be optimal for homoskedastic Gaussian noise. Furthermore, as a byproduct of our theoretical results, we establish the entrywise convergence of HOOI when initialized via diagonal deletion. To further illustrate the utility of our theoretical results, we then consider several concrete statistical inference tasks. First, in the tensor mixed-membership blockmodel, we consider a two-sample test for equality of membership profiles, and we propose a test statistic with consistency under local alternatives that exhibits a power improvement relative to the corresponding matrix test considered in several previous works. Next, we consider simultaneous inference for small collections of entries of the tensor, and we obtain consistent confidence regions. Finally, focusing on the particular case of testing whether entries of the tensor are equal, we propose a consistent test statistic that shows how index overlap results in different asymptotic standard deviations. All of our proposed procedures are fully data-driven, adaptive to noise distribution and signal strength, and do not rely on sample-splitting, and our main results highlight the effect of higher-order structures on estimation relative to the matrix setting. Our theoretical results are demonstrated through numerical simulations.<\/jats:p>","DOI":"10.1214\/25-aos2589","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:50:35Z","timestamp":1782395435000},"page":"1178-1204","source":"Crossref","is-referenced-by-count":0,"title":["Statistical inference for low-rank tensors: Heteroskedasticity, subgaussianity, and applications"],"prefix":"10.1214","volume":"54","author":[{"given":"Joshua","family":"Agterberg","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Illinois Urbana-Champaign"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Anru R.","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Biostatistics & Bioinformatics and Department of Computer Science, Duke University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"27","doi-asserted-by":"publisher","unstructured":"<small>Kolda, T. G.<\/small> and <small>Bader, B. W.<\/small> (2009). Tensor decompositions and applications. <i>SIAM Rev.<\/i> <b>51<\/b> 455\u2013500.","DOI":"10.1137\/07070111X"},{"key":"28","doi-asserted-by":"publisher","unstructured":"<small>Lei, J.<\/small>, <small>Chen, K.<\/small> and <small>Lynch, B.<\/small> (2020). Consistent community detection in multi-layer network data. <i>Biometrika<\/i> <b>107<\/b> 61\u201373.","DOI":"10.1093\/biomet\/asz068"},{"key":"43","doi-asserted-by":"publisher","unstructured":"<small>Zhang, A.<\/small> and <small>Han, R.<\/small> (2019). Optimal sparse singular value decomposition for high-dimensional high-order data. <i>J. Amer. Statist. Assoc.<\/i> <b>114<\/b> 1708\u20131725.","DOI":"10.1080\/01621459.2018.1527227"},{"key":"11","doi-asserted-by":"publisher","unstructured":"<small>Cai, T. T.<\/small> and <small>Guo, Z.<\/small> (2017). Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity. <i>Ann. Statist.<\/i> <b>45<\/b> 615\u2013646.","DOI":"10.1214\/16-AOS1461"},{"key":"16","doi-asserted-by":"publisher","unstructured":"<small>FAN, J.<\/small>, <small>FAN, Y.<\/small>, <small>HAN, X.<\/small> and <small>LV, J.<\/small> (2022). SIMPLE: Statistical inference on membership profiles in large networks. <i>J. R. Stat. Soc. Ser. B. Stat. Methodol.<\/i> <b>84<\/b> 630\u2013653.","DOI":"10.1111\/rssb.12505"},{"key":"12","unstructured":"<small>CHEN, Y.<\/small>, <small>CHI, Y.<\/small>, <small>FAN, J.<\/small> and <small>MA, C.<\/small> (2021). Spectral methods for data science: A statistical perspective. <i>Found. Trends Mach. Learn.<\/i> <b>14<\/b> 566\u2013806."},{"key":"1","doi-asserted-by":"publisher","unstructured":"<small>Abbe, E.<\/small>, <small>Fan, J.<\/small>, <small>Wang, K.<\/small> and <small>Zhong, Y.<\/small> (2020). Entrywise eigenvector analysis of random matrices with low expected rank. <i>Ann. Statist.<\/i> <b>48<\/b> 1452\u20131474.","DOI":"10.1214\/19-AOS1854"},{"key":"22","doi-asserted-by":"publisher","unstructured":"<small>HAO, B.<\/small>, <small>ZHANG, A.<\/small> and <small>CHENG, G.<\/small> (2020). Sparse and low-rank tensor estimation via cubic sketchings. <i>IEEE Trans. Inf. Theory<\/i> <b>66<\/b> 5927\u20135964.","DOI":"10.1109\/TIT.2020.2982499"},{"key":"29","doi-asserted-by":"publisher","unstructured":"<small>LI, L.<\/small> and <small>ZHANG, X.<\/small> (2017). Parsimonious tensor response regression. <i>J. Amer. Statist. Assoc.<\/i> <b>112<\/b> 1131\u20131146.","DOI":"10.1080\/01621459.2016.1193022"},{"key":"13","unstructured":"<small>CHI, E. C.<\/small>, <small>GAINES, B. R.<\/small>, <small>SUN, W. W.<\/small>, <small>ZHOU, H.<\/small> and <small>YANG, J.<\/small> (2020). Provable convex co-clustering of tensors. <i>J. Mach. Learn. Res.<\/i> <b>21<\/b> Paper No. 214, 58."},{"key":"26","doi-asserted-by":"publisher","unstructured":"<small>JING, B.-Y.<\/small>, <small>LI, T.<\/small>, <small>LYU, Z.<\/small> and <small>XIA, D.<\/small> (2021). Community detection on mixture multilayer networks via regularized tensor decomposition. <i>Ann. Statist.<\/i> <b>49<\/b> 3181\u20133205.","DOI":"10.1214\/21-aos2079"},{"key":"45","doi-asserted-by":"publisher","unstructured":"<small>ZHANG, A. R.<\/small>, <small>CAI, T. T.<\/small> and <small>WU, Y.<\/small> (2022). Heteroskedastic PCA: Algorithm, optimality, and applications. <i>Ann. Statist.<\/i> <b>50<\/b> 53\u201380.","DOI":"10.1214\/21-aos2074"},{"key":"47","doi-asserted-by":"crossref","unstructured":"<small>ZHANG, Z.<\/small>, <small>ALLEN, G. I.<\/small>, <small>ZHU, H.<\/small> and <small>DUNSON, D.<\/small> (2019). Tensor network factorizations: Relationships between brain structural connectomes and traits. <i>NeuroImage<\/i> <b>197<\/b> 330\u2013343.","DOI":"10.1016\/j.neuroimage.2019.04.027"},{"key":"44","doi-asserted-by":"publisher","unstructured":"<small>Zhang, A.<\/small> and <small>Xia, D.<\/small> (2018). Tensor SVD: Statistical and computational limits. <i>IEEE Trans. Inf. Theory<\/i> <b>64<\/b> 7311\u20137338.","DOI":"10.1109\/TIT.2018.2841377"},{"key":"37","doi-asserted-by":"publisher","unstructured":"<small>Vershynin, R.<\/small> (2018). <i>High-Dimensional Probability<\/i>: <i>An Introduction with Applications in Data Science<\/i>. <i>Cambridge Series in Statistical and Probabilistic Mathematics<\/i> <b>47<\/b>. Cambridge Univ. Press, Cambridge.","DOI":"10.1017\/9781108231596"},{"key":"36","unstructured":"<small>Richard, E.<\/small> and <small>Montanari, A.<\/small> (2014). A statistical model for tensor PCA. In <i>Advances in Neural Information Processing Systems<\/i> 2897\u20132905."},{"key":"41","doi-asserted-by":"publisher","unstructured":"<small>XIA, D.<\/small>, <small>ZHANG, A. R.<\/small> and <small>ZHOU, Y.<\/small> (2022). Inference for low-rank tensors\u2014no need to debias. <i>Ann. Statist.<\/i> <b>50<\/b> 1220\u20131245.","DOI":"10.1214\/21-aos2146"},{"key":"42","doi-asserted-by":"publisher","unstructured":"<small>YAN, Y.<\/small>, <small>CHEN, Y.<\/small> and <small>FAN, J.<\/small> (2024). Inference for heteroskedastic PCA with missing data. <i>Ann. Statist.<\/i> <b>52<\/b> 729\u2013756.","DOI":"10.1214\/24-aos2366"},{"key":"49","doi-asserted-by":"publisher","unstructured":"<small>ZHOU, Y.<\/small>, <small>ZHANG, A. R.<\/small>, <small>ZHENG, L.<\/small> and <small>WANG, Y.<\/small> (2022). Optimal high-order tensor SVD via tensor-train orthogonal iteration. <i>IEEE Trans. Inf. Theory<\/i> <b>68<\/b> 3991\u20134019.","DOI":"10.1109\/tit.2022.3152733"},{"key":"34","doi-asserted-by":"publisher","unstructured":"<small>LYU, Z.<\/small>, <small>XIA, D.<\/small> and <small>ZHANG, Y.<\/small> (2023). Latent space model for higher-order networks and generalized tensor decomposition. <i>J. Comput. Graph. Statist.<\/i> <b>32<\/b> 1320\u20131336.","DOI":"10.1080\/10618600.2022.2164289"},{"key":"23","doi-asserted-by":"publisher","unstructured":"<small>HU, J.<\/small> and <small>WANG, M.<\/small> (2023). Multiway spherical clustering via degree-corrected tensor block models. <i>IEEE Trans. Inf. Theory<\/i> <b>69<\/b> 3880\u20133919.","DOI":"10.1109\/tit.2023.3239521"},{"key":"15","doi-asserted-by":"publisher","unstructured":"<small>DU, X.<\/small> and <small>TANG, M.<\/small> (2023). Hypothesis testing for equality of latent positions in random graphs. <i>Bernoulli<\/i> <b>29<\/b> 3221\u20133254.","DOI":"10.3150\/22-bej1581"},{"key":"20","doi-asserted-by":"publisher","unstructured":"<small>HAN, X.<\/small>, <small>YANG, Q.<\/small> and <small>FAN, Y.<\/small> (2023). Universal rank inference via residual subsampling with application to large networks. <i>Ann. Statist.<\/i> <b>51<\/b> 1109\u20131133.","DOI":"10.1214\/23-aos2282"},{"key":"33","doi-asserted-by":"publisher","unstructured":"<small>LYU, Z.<\/small> and <small>XIA, D.<\/small> (2026). Optimal clustering by Lloyd\u2019s algorithm for low-rank mixture model. <i>J. R. Stat. Soc. Ser. B. Stat. Methodol.<\/i> <b>88<\/b> 43\u201365.","DOI":"10.1093\/jrsssb\/qkaf041"},{"key":"3","doi-asserted-by":"publisher","unstructured":"<small>AGTERBERG, J.<\/small> and <small>ZHANG, A. R.<\/small> (2025). Estimating higher-order mixed memberships via the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" id=\"eq_3\"><msub><mrow><mi>\u2113<\/mi><\/mrow><mrow><mn>2<\/mn><mo>,<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><\/math> tensor perturbation bound. <i>J. Amer. Statist. Assoc.<\/i> <b>120<\/b> 1214\u20131224.","DOI":"10.1080\/01621459.2024.2404265"},{"key":"4","doi-asserted-by":"crossref","unstructured":"<small>AGTERBERG, J.<\/small> and <small>ZHANG, A. R<\/small> (2026). Supplement to \u201cStatistical inference for low-rank tensors: Heteroskedasticity, subgaussianity, and applications.\u201d <a href=\"https:\/\/doi.org\/10.1214\/25-AOS2589SUPP\">https:\/\/doi.org\/10.1214\/25-AOS2589SUPP<\/a>","DOI":"10.1214\/25-AOS2589SUPP"},{"key":"7","doi-asserted-by":"publisher","unstructured":"<small>CAI, C.<\/small>, <small>LI, G.<\/small>, <small>CHI, Y.<\/small>, <small>POOR, H. V.<\/small> and <small>CHEN, Y.<\/small> (2021). Subspace estimation from unbalanced and incomplete data matrices: <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" id=\"eq_4\"><msub><mrow><mi>\u2113<\/mi><\/mrow><mrow><mn>2<\/mn><mo>,<\/mo><mi>\u221e<\/mi><\/mrow><\/msub><\/math> statistical guarantees. <i>Ann. Statist.<\/i> <b>49<\/b> 944\u2013967.","DOI":"10.1214\/20-aos1986"},{"key":"10","unstructured":"<small>CAI, J.-F.<\/small>, <small>LI, J.<\/small> and <small>XIA, D.<\/small> (2022). Provable tensor-train format tensor completion by Riemannian optimization. <i>J. Mach. Learn. Res.<\/i> <b>23<\/b> Paper No. [123], 77."},{"key":"14","doi-asserted-by":"publisher","unstructured":"<small>DE LATHAUWER, L.<\/small>, <small>DE MOOR, B.<\/small> and <small>VANDEWALLE, J.<\/small> (2000). On the best rank-1 and rank-<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" id=\"eq_5\"><mo fence=\"true\" stretchy=\"false\">(<\/mo><msub><mrow><mi mathvariant=\"italic\">R<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo>,<\/mo><msub><mrow><mi mathvariant=\"italic\">R<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mrow><mi mathvariant=\"italic\">R<\/mi><\/mrow><mrow><mi mathvariant=\"italic\">N<\/mi><\/mrow><\/msub><mo fence=\"true\" stretchy=\"false\">)<\/mo><\/math> approximation of higher-order tensors. <i>SIAM J. Matrix Anal. Appl.<\/i> <b>21<\/b> 1324\u20131342.","DOI":"10.1137\/S0895479898346995"},{"key":"18","doi-asserted-by":"publisher","unstructured":"<small>HAN, R.<\/small>, <small>SHI, P.<\/small> and <small>ZHANG, A. R.<\/small> (2024). Guaranteed functional tensor singular value decomposition. <i>J. Amer. Statist. Assoc.<\/i> <b>119<\/b> 995\u20131007.","DOI":"10.1080\/01621459.2022.2153689"},{"key":"24","unstructured":"<small>HUANG, J.<\/small>, <small>HUANG, D. Z.<\/small>, <small>YANG, Q.<\/small> and <small>CHENG, G.<\/small> (2022). Power iteration for tensor PCA. <i>J. Mach. Learn. Res.<\/i> <b>23<\/b> Paper No. [128], 47."},{"key":"30","unstructured":"<small>LUO, Y.<\/small>, <small>RASKUTTI, G.<\/small>, <small>YUAN, M.<\/small> and <small>ZHANG, A. R.<\/small> (2021). A sharp blockwise tensor perturbation bound for orthogonal iteration. <i>J. Mach. Learn. Res.<\/i> <b>22<\/b> Paper No. 179, 48."},{"key":"35","unstructured":"<small>MARTINO, C.<\/small>, <small>SHENHAV, L.<\/small>, <small>MAROTZ, C. A.<\/small>, <small>ARMSTRONG, G.<\/small>, <small>MCDONALD, D.<\/small>, <small>V\u00c1ZQUEZ-BAEZA, Y.<\/small>, <small>MORTON, J. T.<\/small>, <small>JIANG, L.<\/small>, <small>GLORIA DOMINGUEZ-BELLO, M.<\/small> et al. (2021). Context-aware dimensionality reduction deconvolutes gut microbial community dynamics. <i>Nat. Biotechnol.<\/i> <b>39<\/b> 165\u2013168."},{"key":"39","unstructured":"<small>WANG, M.<\/small> and <small>ZENG, Y.<\/small> (2019). Multiway clustering via tensor block models. In <i>Advances in Neural Information Processing Systems<\/i> <b>32<\/b>. Curran Associates, Red Hook."},{"key":"40","unstructured":"<small>WU, T.<\/small>, <small>BENSON, A. R.<\/small> and <small>GLEICH, D. F.<\/small> (2016). General tensor spectral co-clustering for higher-order data. In <i>Advances in Neural Information Processing Systems<\/i> <b>29<\/b>. Curran Associates, Red Hook."},{"key":"46","doi-asserted-by":"crossref","unstructured":"<small>ZHANG, C.<\/small>, <small>HAN, R.<\/small>, <small>ZHANG, A. R.<\/small> and <small>VOYLES, P. M.<\/small> (2020). Denoising atomic resolution 4D scanning transmission electron microscopy data with tensor singular value decomposition. <i>Ultramicroscopy<\/i> <b>219<\/b> 113123.","DOI":"10.1016\/j.ultramic.2020.113123"},{"key":"48","doi-asserted-by":"publisher","unstructured":"<small>ZHOU, Y.<\/small> and <small>CHEN, Y.<\/small> (2025). Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA. <i>Ann. Statist.<\/i> <b>53<\/b> 91\u2013116.","DOI":"10.1214\/24-aos2456"},{"key":"50","doi-asserted-by":"publisher","unstructured":"<small>ZHU, M.<\/small> and <small>GHODSI, A.<\/small> (2006). Automatic dimensionality selection from the scree plot via the use of profile likelihood. <i>Comput. Statist. Data Anal.<\/i> <b>51<\/b> 918\u2013930.","DOI":"10.1016\/j.csda.2005.09.010"},{"key":"19","doi-asserted-by":"publisher","unstructured":"<small>HAN, R.<\/small>, <small>WILLETT, R.<\/small> and <small>ZHANG, A. R.<\/small> (2022). An optimal statistical and computational framework for generalized tensor estimation. <i>Ann. Statist.<\/i> <b>50<\/b> 1\u201329.","DOI":"10.1214\/21-AOS2061"},{"key":"31","doi-asserted-by":"publisher","unstructured":"<small>LUO, Y.<\/small> and <small>ZHANG, A. R.<\/small> (2022). Tensor clustering with planted structures: Statistical optimality and computational limits. <i>Ann. Statist.<\/i> <b>50<\/b> 584\u2013613.","DOI":"10.1214\/21-aos2123"},{"key":"25","unstructured":"<small>JIN, J.<\/small>, <small>KE, Z. T.<\/small>, <small>LUO, S.<\/small> and <small>WANG, M.<\/small> (2023). Optimal Estimation of the Number of Network Communities. <i>J. Amer. Statist. Assoc.<\/i> <b>118<\/b> 2101\u20132116."},{"key":"21","doi-asserted-by":"publisher","unstructured":"<small>HAN, Y.<\/small> and <small>ZHANG, C.-H.<\/small> (2023). Tensor principal component analysis in high dimensional CP models. <i>IEEE Trans. Inf. Theory<\/i> <b>69<\/b> 1147\u20131167.","DOI":"10.1109\/tit.2022.3203972"},{"key":"2","doi-asserted-by":"crossref","unstructured":"<small>AGTERBERG, J.<\/small>, <small>LUBBERTS, Z.<\/small> and <small>PRIEBE, C. E.<\/small> (2022). Entrywise estimation of singular vectors of low-rank matrices with heteroskedasticity and dependence. <i>IEEE Trans. Inf. Theory<\/i> <b>68<\/b> 4618\u20134650.","DOI":"10.1109\/TIT.2022.3159085"},{"key":"8","doi-asserted-by":"crossref","unstructured":"<small>CAI, C.<\/small>, <small>LI, G.<\/small>, <small>POOR, H. V.<\/small> and <small>CHEN, Y.<\/small> (2022). Nonconvex low-rank tensor completion from noisy data. <i>Oper. Res.<\/i> <b>70<\/b> 1219\u20131237.","DOI":"10.1287\/opre.2021.2106"},{"key":"9","doi-asserted-by":"crossref","unstructured":"<small>CAI, C.<\/small>, <small>POOR, H. V.<\/small> and <small>CHEN, Y.<\/small> (2023). Uncertainty quantification for nonconvex tensor completion: Confidence intervals, heteroscedasticity and optimality. <i>IEEE Trans. Inf. Theory<\/i> <b>69<\/b> 407\u2013452.","DOI":"10.1109\/TIT.2022.3205781"},{"key":"17","doi-asserted-by":"crossref","unstructured":"<small>HAN, R.<\/small>, <small>LUO, Y.<\/small>, <small>WANG, M.<\/small> and <small>ZHANG, A. R.<\/small> (2022). Exact clustering in tensor block model: Statistical optimality and computational limit. <i>J. R. Stat. Soc. Ser. B. Stat. Methodol.<\/i> <b>84<\/b> 1666\u20131698.","DOI":"10.1111\/rssb.12547"},{"key":"32","doi-asserted-by":"publisher","unstructured":"<small>LYU, Z.<\/small> and <small>XIA, D.<\/small> (2023). Optimal estimation and computational limit of low-rank Gaussian mixtures. <i>Ann. Statist.<\/i> <b>51<\/b> 646\u2013667.","DOI":"10.1214\/23-aos2264"},{"key":"38","unstructured":"<small>WANG, H.<\/small>, <small>CHEN, J.<\/small> and <small>WEI, K.<\/small> (2023). Implicit regularization and entrywise convergence of Riemannian optimization for low Tucker-rank tensor completion. <i>J. Mach. Learn. Res.<\/i> <b>24<\/b> Paper No. [347], 84."},{"key":"5","doi-asserted-by":"publisher","unstructured":"<small>AUDDY, A.<\/small> and <small>YUAN, M.<\/small> (2022). On estimating rank-one spiked tensors in the presence of heavy tailed errors. <i>IEEE Trans. Inf. Theory<\/i> <b>68<\/b> 8053\u20138075.","DOI":"10.1109\/tit.2022.3191883"},{"key":"6","doi-asserted-by":"publisher","unstructured":"<small>AUDDY, A.<\/small> and <small>YUAN, M.<\/small> (2023). Perturbation bounds for (nearly) orthogonally decomposable tensors with statistical applications. <i>Inf. Inference<\/i> <b>12<\/b> 1044\u20131072.","DOI":"10.1093\/imaiai\/iaac033"}],"container-title":["The Annals of Statistics"],"original-title":[],"deposited":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:19:15Z","timestamp":1791321555000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-54\/issue-3\/Statistical-inference-for-low-rank-tensors--Heteroskedasticity-subgaussianity-and\/10.1214\/25-AOS2589.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,1]]},"references-count":50,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,6,1]]}},"URL":"https:\/\/doi.org\/10.1214\/25-aos2589","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,6,1]]}}}