Tensor Elliptical Graphic Model

Jixuan Liu; Le Zhou; Long Feng; Zhaojun Wang; Zhengke Lu

REVIEW 3 major objections 5 minor 51 references

Tensor Elliptical Graphic Model

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that spatial-sign-transformed tensor data yield mode-wise precision matrix estimators for tensor elliptical graphical models at the same convergence rate as Gaussian tensor graphical models, without iterative updates…

desk verdict A useful tensor extension of spatial-sign estimation, but the main theorem ignores the model's own location parameter and the rate is unsupported as written. read the letter →

arxiv 2508.00333 v1 pith:3AP3K6AO submitted 2025-08-01 stat.ME

Jixuan Liu , Zhengke Lu , Le Zhou , Long Feng , Zhaojun Wang This is my paper · ORCID

classification stat.ME MSC 62H1262F35

keywords tensorellipticaldistributionspatialsignprecisionmatrixestimationgraphicalmodelheavy-taileddatalassohigh-dimensionalstatisticsKronecker-structuredcovariance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Most tensor graphical model estimators assume tensor normality. This paper extends the setting to tensor elliptical distributions, which include heavy-tailed members such as tensor $t$-distributions. It proposes a spatial-sign-based estimator that processes each mode separately and runs the graphical lasso once per mode, so no iterative updates across modes are needed. The central result is that each mode-wise precision matrix estimate achieves the same Frobenius-norm convergence rate as existing tensor Gaussian estimators, while allowing heavy-tailed data. If correct, this gives practitioners a heavy-tail-aware tensor graphical model recovery method at no asymptotic cost in rate.

What carries the argument

The central object is the spatial-sign covariance matrix: a covariance matrix computed from unit-norm direction vectors $X/\|X\|$ instead of raw observations, which is insensitive to the radial tail of the distribution. The paper exploits the fact that for tensor elliptical distributions this matrix is, up to a constant, a consistent proxy for the true Kronecker-structured covariance when the ambient dimension $p^*$ grows. Each mode's precision matrix is then recovered by running the graphical lasso on the corresponding mode-wise proxy, and the proof of Theorem 1 uses this proportionality to transfer concentration bounds from sign-transformed data to the true precision matrix.

What would settle it

Simulate tensor $t$-distributed data with known mode-wise precision matrices and tail parameter $\nu=3$, choose $p_k$ and $n$ so that Theorem 1's regime holds, and check whether the empirical Frobenius error of $\hat\Omega_k$ tracks the stated rate as $p^*$ grows. A simpler decisive check is to estimate the ratio between $E[XX^\top/\|X\|^2]$ and $\Sigma^*$ directly: if this ratio depends on $\nu$ rather than being a constant as $p^*\to\infty$, the spatial-sign proxy fails and the bound should not hold.

Watch

Extended reading notes

Core claim

The paper claims that under tensor elliptical sampling, for each mode $k$ the sparse spatial-sign estimator $\hat{\Omega}_k$ satisfies $$ \|\hat{\$\Omega$}_k - \Omega_k^*\|_F = O_p\left(\sqrt{\frac{(p_k+s_k)p_k \log p_k}{n p^*}}\right) + O\left(\frac{\sqrt{p_k+s_k}}{$p_k^{2}$}\right), $$ where $p^*=\prod_j p_j$, $s_k$ counts the nonzero off-diagonal entries of $\Omega_k^*$, and $s_k=O(p_k)$. The first term is the statistical error and the second is the approximation bias inherited from the spatial-sign proxy. The paper argues this matches the established rate for Gaussian tensor graphical models and replaces the normality requirement with the tensor elliptical family. It also emphasizes that the estimator is one-step and mode-wise: no cyclic or alternating optimization across modes is needed.

Load-bearing premise

The load-bearing premise is that for tensor elliptical distributions, as the dimension grows, the spatial-sign covariance matrix equals the true covariance matrix times a constant, so the sign-transformed data carry the same conditional-dependence information as the original data. If this proportionality holds only under extra tail or eigenvalue conditions, the stated rates would not follow.

Editorial extensions

If this is right

The convergence rate available for Gaussian tensor graphical models becomes available for every tensor elliptical distribution, including heavy-tailed members such as tensor $t$ distributions.
Because each mode is handled separately and only once, practitioners avoid iterative mode-updating schemes and their computational cost.
The zero pattern of each mode's precision matrix remains identifiable under heavy tails, so conditional-independence structure can be recovered without normality.
The paper's simulations and EEG application indicate the estimator stays competitive with mean-based methods on normal data and improves on them under heavy tails.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The single-pass, per-mode structure makes the estimator naturally parallelisable across the $K$ modes, and it could tolerate different tail behavior in different modes without modification.
The rate contains a bias term $O(\sqrt{p_k+s_k}/p_k^2)$, so the advertised rate is a high-dimensional asymptotic statement; small mode dimensions may not show the same behavior in finite samples.
The unknown scale relating the spatial-sign covariance to the true covariance means the method identifies the precision matrices up to a constant, which is enough for graph recovery but not for scale-dependent quantities.
The same spatial-sign covariance proxy could be dropped into other tensor pipelines that consume covariance or precision estimates, such as tensor classifier or discriminant analysis, where heavy-tailed neuroimaging or genomic data are common.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version. The paper's genuine contribution is to push spatial-sign precision estimation from vector and matrix settings to tensor elliptical graphical models, with one-step mode-wise computation and an explicit Frobenius rate that would match the Gaussian tensor rate. That is a real extension and, if it works, practically useful.

What it does well: the estimator is simple, avoids iterative updates, and the heavy-tailed motivation is legitimate for neuroimaging-type data. The theoretical rate claim, with the sparsity-adjusted term, looks plausible at a glance, and the paper honestly builds on prior spatial-sign work rather than overselling novelty.

Now the soft spots. The biggest one is the location parameter. The model in Section 2.1 defines X = mu + v Sigma^{1/2} u, but Algorithm 1, the assumptions, and Theorem 1 never mention mu, and there is no centering step or robust location estimator. The key proxy identity behind Theorem 1, invoked as 'Proof of Equation (6.5)', is that E[S(X)S(X)^T] is asymptotically proportional to the covariance. That identity holds for centered spatial signs, not for raw signs applied to non-centered data. With mu nonzero, the raw sign's expectation is contaminated by mu, and the theorem has no term for this and no condition on mu. As written, Theorem 1 is unsupported for the very model the paper introduces. This is fixable: either assume centered data explicitly or estimate mu and carry the error through the rate. But it is a load-bearing gap, not a cosmetic one.

There are other issues. The arXiv text is partly garbled—Assumptions 1–3 and Theorem 1 are hard to read, and Section 3 simulations are missing from the version I saw, so I could not check the 'extensive simulations' claim. The rate also contains an O(1/p_k^2) term whose role is not explained. The real-data analysis is qualitative, fine for an illustration but not evidence for the rate.

Where does that leave us? The central idea is credible and the direction is worth pursuing. The location gap is the kind of thing a good referee would catch, and it can be fixed. I would not desk-reject this; I would send it to a careful referee and expect a major revision that states the centering assumption (or estimates mu) and makes the proof of the proxy identity fully visible.

Who's it for? Anyone working on robust high-dimensional tensor models. I wouldn't cite it until the location issue is resolved, but I'd bring it to a reading group to discuss the gap.

Referee Report

3 major / 5 minor

Summary. The paper proposes a spatial-sign based estimator for the precision matrices of a tensor elliptical graphical model. For each mode k, the estimator computes a normalized spatial-sign covariance matrix, applies the graphical lasso, and normalizes the result to unit Frobenius norm. The central theoretical claim is that under eigenvalue and sparsity assumptions, each mode-wise precision matrix estimator attains an error rate comparable to Gaussian tensor graphical model estimators, without iterative updates across modes and under heavy-tailed elliptical distributions. The paper also reports simulations and a real-data application to EEG data.

Significance. The proposed method is simple, non-iterative, and free of fitted constants in the error bound, which are real strengths. If the convergence-rate result is correct, it would extend robust precision matrix estimation to tensor elliptical distributions at a rate comparable to Gaussian methods, a useful step beyond the normality assumption that dominates the tensor graphical model literature. However, the two load-bearing issues listed below—the absence of a centering or location assumption/estimation step, and the missing proof of the key spatial-sign covariance proxy identity—must be resolved before the significance claim can be accepted. The stated rate also needs reconciliation with the rate shown in the appendix proof.

major comments (3)

[Section 2.1, Algorithm 1, Theorem 1] The model is defined in Section 2.1 as x = vec(X) = μ + v(Σ*)^{1/2}u with a location parameter μ, but Algorithm 1, Assumptions 1–3, and Theorem 1 do not assume μ = 0, do not center the data, and do not estimate μ. The spatial-sign proxy identity invoked in the proof of Equation (6.5), E[S(X)S(X)^T] ≈ constant × Σ*, holds for the centered spatial sign S(X − μ), not for the raw spatial sign S(X). For μ ≠ 0, the expectation of S(X)S(X)^T contains μ-dependent terms that do not vanish as p* grows unless μ is negligible in the relevant norm, and the theorem's rate contains no location-estimation term. As written, Theorem 1 is therefore unsupported for the model actually introduced; the authors must either add an explicit center-at-zero assumption or estimate μ and account for that estimation error in the rate.
[Appendix, Proof of Equation (6.5)] The appendix contains a heading 'Proof of Equation (6.5)' but no proof follows; the captured text jumps directly to the next page. Equation (6.5) is the identity that the spatial-sign covariance matrix is asymptotically a scalar multiple of the true covariance matrix, which the abstract states as the motivation for the method and which Theorem 1 relies on. Without a proof of this identity, with precise conditions on p*, n, and the distribution of the radius v, the main theorem is not established. Please supply the proof or a precise reference to a theorem that covers the tensor elliptical case.
[Section 2.3 (Theorem 1) and Appendix proof] The rate stated in Theorem 1 is O_p( sqrt((p_k+s_k)p_k log p_k/(n p*)) ) + O( sqrt(p_k+s_k)/p_k^2 ), which under s_k = O(p_k) is O_p( p_k sqrt(log p_k/(n p*)) ) + O(p_k^{-3/2}). The proof's final display for the Frobenius norm error gives O_p( sqrt(p_k log p_k/(n p*)) ) + O(1/p_k). These two rates differ by a factor of sqrt(p_k) in the leading term, and the bias terms also differ. If the proof's derivation is correct, Theorem 1's statement is too slow; if Theorem 1's statement is intended, the proof must be corrected. The authors should also state explicitly which Gaussian tensor graphical model rate they are comparing with, since the claimed 'matches the existing rate under normality' is not quantified in the text.

minor comments (5)

[Title/Abstract] The title uses 'Graphic Model' while the text uses 'Graphical Model'; unify the terminology.
[Section 2.3] The typeset statement of Assumptions 1–3 and Theorem 1 is garbled in the provided text; if this reflects the actual PDF, those statements need to be retypeset so that the conditions and the rate are legible.
[Algorithm 1] The return statement in Algorithm 1 refers to Ω̂(t)_1,...,Ω̂(t)_K with an undefined superscript t; remove or define it.
[Appendix proof fragment] In the proof of Theorem 1, the final display appears to read 'O(1−p*)' where 'O(1/p*)' is presumably intended; please check and correct.
[Section 2.3] When claiming that the rate 'matches the existing rate under normality', the authors should cite the specific rate from Min et al. (2022) or another reference and show the comparison explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the core proxy identity is proven as a lemma from the tensor-elliptical model, and the rate in Theorem 1 follows from concentration bounds rather than from fitted inputs or self-cited conclusions.

full rationale

The paper's derivation is self-contained for its central claim. The key input is the spatial-sign covariance proxy, stated in the abstract and used as Equation (6.5) in the proof of Theorem 1. The text labels the derivation as 'Proof of Equation (6.5)', indicating that the identity is proved from the tensor elliptical model (concentration of quadratic forms of the uniform spherical vector under Assumptions 1-2), not imported as a black-box conclusion equal to the theorem. The rate in Theorem 1 is an explicit function of n, p_k, s_k and p*, with no fitted constants renamed as predictions; the normalization step ||Ω^^k||_F=1 defines the scale of the estimator, and any identifiability issue would be a modeling/correctness concern rather than circularity. The paper cites the authors' own prior work (Lu and Feng, 2025) for the existing spatial-sign precision estimator and for robustness motivation, but the tensor extension, the Kronecker-structured mode-wise estimation, and the concentration proof are developed in the present paper; the citations are contextual, not the load-bearing justification for Theorem 1. The skeptic's point about the location parameter μ introduced in Section 2.1 is a potential missing assumption (the proxy identity needs centered data), but that is a correctness gap, not an equivalence of the result with its inputs by construction. Accordingly, no circular step is identified; the score reflects only the presence of minor non-load-bearing self-citations.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. It relies on a distributional family, standard concentration inequalities, sparsity assumptions, and a stated but unproven proportionality property of spatial sign covariance under tensor elliptical distributions. The only data-fitted quantities are the per-mode tuning parameters λ_k.

free parameters (1)

Tuning parameters λ_k (k = 1,...,K) = chosen by cross-validation in practice
Assumption 3 specifies a scaling interval for λ_k, and Algorithm 1 requires an input λ_k; in the EEG application λ_k is selected by 5-fold cross-validation. The theoretical rate depends on λ_k lying in this oracle interval.

assumptions (5)

domain assumption Tensor elliptical distribution assumption: x = μ + v (Σ*)^{1/2} u with v independent of uniform u, and density generator g.
The model posits this family for the observed tensors; all results are conditional on this distributional assumption.
domain assumption Spatial-sign covariance proportionality: E[u(X)u(X)^T] is (up to a constant) the true covariance when the dimension tends to infinity under tensor elliptical distribution.
Stated in the abstract and used as the motivation for the estimator in Section 2.2 and in the proof of Theorem 1; the visible text does not provide a derivation or citation for this key fact.
domain assumption Bounded eigenvalues and L1 norms (Assumptions 1 and 2): 0 < C1 ≤ λ_min(Σ*_k) ≤ λ_max(Σ*_k) ≤ 1/C1 and max(||Σ*||_L1, ||(Σ*)^{-1}||_L1) ≤ T.
Theorem 1 requires these conditions on the true covariance matrices for all modes.
domain assumption Sparsity condition s_k = O(p_k) for each mode k.
The rate expression simplifies under the assumption that the number of nonzero off-diagonal entries in Ω*_k grows at most linearly in p_k.
standard math Sub-Gaussian and sub-exponential concentration inequalities for sums of spatial sign outer products.
The proof uses concentration results from Vershynin (2018), which are standard tools for random vectors with bounded norms.

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Cite this review

Pith. "Pith review of Tensor Elliptical Graphic Model." pith.science (2026). https://pith.science/paper/3AP3K6AO

@misc{pith2026250800333,
  author       = {Pith},
  title        = {Pith review of: Tensor Elliptical Graphic Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AP3K6AO}},
  note         = {Machine review of arXiv:2508.00333}
}

read the original abstract

We address the problem of robust estimation of sparse high dimensional tensor elliptical graphical model. Most of the research focus on tensor graphical model under normality. To extend the tensor graphical model to more heavy-tailed scenarios, motivated by the fact that up to a constant, the spatial-sign covariance matrix can approximate the true covariance matrix when the dimension turns to infinity under tensor elliptical distribution, we proposed a spatial-sign-based estimator to robustly estimate tensor elliptical graphical model, the rate of which matches the existing rate under normality for a wider family of distribution, i.e. elliptical distribution. We also conducted extensive simulations and real data applications to illustrate the practical utility of the proposed methods, especially under heavy-tailed distribution.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 44 canonical work pages

[1]

Arashi, M. (2023). Some theoretical results on the tensor elliptical distribution. Journal of Data Science and Modeling\/ 1\/ (2), 29--41

work page 2023
[2]

Shibata, and M

Baba, K., R. Shibata, and M. Sibuya (2004). Partial correlation and conditional correlation as measures of conditional independence. Australian & New Zealand Journal of Statistics\/ 46\/ (4), 657--664

work page 2004
[3]

Bickel, P. J. and E. Levina (2008). Covariance regularization by thresholding. The Annals of Statistics\/ 36\/ (6), 2577--2604

work page 2008
[4]

Liu, and X

Cai, T., W. Liu, and X. Luo (2011). A constrained l1 minimization approach to sparse precision matrix estimation. Journal of the American Statistical Association\/ 106\/ (494), 594--607

work page 2011
[5]

Cai, T. T., W. Liu, and H. H. Zhou (2016). Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation. The Annals of Statistics\/ 44\/ (2), 455--488

work page 2016
[6]

High-Dimensional Tensor Classification with CP Low-Rank Discriminant Structure

Chen, E., Y. Han, and J. Li (2024). High-dimensional tensor classification with cp low-rank discriminant structure. arXiv preprint arXiv:2409.14397\/

work page Pith review arXiv 2024
[7]

Chen, E. Y. and J. Fan (2023). Statistical inference for high-dimensional matrix-variate factor models. Journal of the American Statistical Association\/ 118\/ (542), 1038--1055

work page 2023
[8]

Statistical Inference for Ultrahigh Dimensional Location Parameter Based on Spatial Median

Cheng, G., L. Peng, and C. Zou (2023). Statistical inference for ultrahigh dimensional location parameter based on spatial median. arXiv preprint arXiv:2301.03126\/

work page Pith review arXiv 2023

Show all 51 references

[9]

Feng, and Y

Fan, J., Y. Feng, and Y. Wu (2009). Network exploration via the adaptive lasso and scad penalties. The Annals of Applied Statistics\/ 3\/ (2), 521

2009
[10]

Fang, K. W. (2018). Symmetric multivariate and related distributions . Chapman and Hall/CRC

2018
[11]

Zou, and Z

Feng, L., C. Zou, and Z. Wang (2016). Multivariate-sign-based high-dimensional tests for the two-sample location problem. Journal of the American Statistical Association\/ 111\/ (514), 721--735

2016
[12]

Hastie, and R

Friedman, J., T. Hastie, and R. Tibshirani (2008). Sparse inverse covariance estimation with the graphical lasso. Biostatistics\/ 9\/ (3), 432--441

2008
[13]

Han, R., Y. Luo, M. Wang, and A. R. Zhang (2022). Exact clustering in tensor block model: Statistical optimality and computational limit. Journal of the Royal Statistical Society Series B: Statistical Methodology\/ 84\/ (5), 1666--1698

2022
[14]

and C.-H

Han, Y. and C.-H. Zhang (2023). The tucker low-rank classification on tensor data. IEEE Transactions on Information Theory\/ 69\/ (2), 1147–1167

2023
[15]

Hao, B., B. Wang, P. Wang, J. Zhang, J. Yang, and W. W. Sun (2021). Sparse tensor additive regression. Journal of machine learning research\/ 22\/ (64), 1--43

2021
[16]

He, S., J. Yin, H. Li, and X. Wang (2014). Graphical model selection and estimation for high dimensional tensor data. Journal of Multivariate Analysis\/ 128 , 165--185

2014
[17]

Vinuela, A

Hore, V., A. Vinuela, A. Buil, J. Knight, M. I. McCarthy, K. Small, and J. Marchini (2016). Tensor decomposition for multiple-tissue gene expression experiments. Nature genetics\/ 48\/ (9), 1094--1100

2016
[18]

Mai, and H

Hou, X., Q. Mai, and H. Zou (2024). Tensor mixture discriminant analysis with applications to sensor array data analysis. The Annals of Applied Atatistics\/ 18\/ (1), 626--641

2024
[19]

Karahan, E., P. A. Rojas-Lopez, M. L. Bringas-Vega, P. A. Vald \'e s-Hern \'a ndez, and P. A. Valdes-Sosa (2015). Tensor analysis and fusion of multimodal brain images. Proceedings of the IEEE\/ 103\/ (9), 1531--1559

2015
[20]

Kolda, T. G. and B. W. Bader (2009). Tensor decompositions and applications. SIAM review\/ 51\/ (3), 455--500

2009
[21]

Li, B., M. K. Kim, and N. Altman (2010). On dimension folding of matrix-or array-valued statistical objects. Ann. Statist.\/ 38\/ (1), 1094--1121

2010
[22]

Mai, and X

Li, J., Q. Mai, and X. Zhang (2025). The tucker low-rank classification on tensor data. Statistica Sinica\/ 35 , 1111--1132

2025
[23]

Li, L. and X. Zhang (2017). Parsimonious tensor response regression. Journal of the American Statistical Association\/ 112\/ (519), 1131--1146

2017
[24]

Lu, Z. and L. Feng (2025). Robust sparse precision matrix estimation and its application. arXiv preprint arXiv:2503.03575\/

2025 arXiv
[25]

Luo, Y. and A. R. Zhang (2022). Tensor clustering with planted structures: Statistical optimality and computational limits. The Annals of Statistics\/ 50 , 584–613

2022
[26]

Luo, Y. and A. R. Zhang (2024). Tensor-on-tensor regression: Riemannian optimization, over-parameterization, statistical-computational gap and their interplay. The Annals of Statistics\/ 52\/ (6), 2583–2612

2024
[27]

Lyu, T., E. F. Lock, and L. E. Eberly (2017). Discriminating sample groups with multi-way data. Biostatistics\/ 18\/ (3), 434--450

2017
[28]

Lyu, X., W. W. Sun, Z. Wang, H. Liu, J. Yang, and G. Cheng (2019). Tensor graphical model: Non-convex optimization and statistical inference. IEEE transactions on pattern analysis and machine intelligence\/ 42\/ (8), 2024--2037

2019
[29]

Zhang, Y

Mai, Q., X. Zhang, Y. Pan, and K. Deng (2022). A doubly enhanced em algorithm for model-based tensor clustering. Journal of the American Statistical Association\/ 117\/ (540), 2120--2134

2022
[30]

Mai, and J

Min, K., Q. Mai, and J. Li (2023). Optimality in high-dimensional tensor discriminant analysis. Pattern Recognition\/ 143 , 109803

2023
[31]

Mai, and X

Min, K., Q. Mai, and X. Zhang (2022). Fast and separable estimation in high-dimensional tensor gaussian graphical models. Journal of Computational and Graphical Statistics\/ 31\/ (1), 294--300

2022
[32]

Oja, H. (2010). Multivariate nonparametric methods with R: an approach based on spatial signs and ranks . Springer Science & Business Media

2010
[33]

Mai, and X

Pan, Y., Q. Mai, and X. Zhang (2019). Covariate-adjusted tensor classification in high dimensions. Journal of the American Statistical Association\/ 114\/ (527), 1305--1319

2019
[34]

Ravikumar, P., M. J. Wainwright, G. Raskutti, and B. Yu (2011). High-dimensional covariance estimation by minimizing l1-penalized log-determinant divergence. Electronic Journal of Statistics\/ 5 , 935

2011
[35]

Sun, W., Z. Wang, H. Liu, and G. Cheng (2015). Non-convex statistical optimization for sparse tensor graphical model. Advances in neural information processing systems\/ 28 , 1081--1089

2015
[36]

Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B: Statistical Methodology\/ 58\/ (1), 267--288

1996
[37]

Vershynin, R. (2018). High-dimensional probability: An introduction with applications in data science , Volume 47. Cambridge university press

2018
[38]

Wang, B., L. Zhou, J. Yang, and Q. Mai (2024). Density-convoluted tensor support vector machines. Statistics and Its Interface\/ 17\/ (2), 231--247

2024
[39]

Wang, and X

Wang, N., W. Wang, and X. Zhang (2024). Parsimonious tensor discriminant analysis. Statistica Sinica\/

2024
[40]

Wang, N. and X. Zhang (2024). Robust and covariance-assisted tensor response regression. Statistics and Its Interface\/ 17\/ (2), 291--303

2024
[41]

Zhang, and B

Wang, N., X. Zhang, and B. Li (2022). Likelihood-based dimension folding for tensor data. Statistica Sinica\/ 32 , 2405--2429

2022
[42]

Zhang, and Q

Wang, N., X. Zhang, and Q. Mai (2023). High-dimensional tensor response regression using the t-distribution. arXiv preprint arXiv:2306.12125\/

2023 arXiv
[43]

Zhang, and Q

Xu, P., T. Zhang, and Q. Gu (2017). Efficient algorithm for sparse tensor-variate gaussian graphical models via gradient descent. In Artificial Intelligence and Statistics , pp.\ 923--932. PMLR

2017
[44]

Zhang, A. R. and R. Han (2019). Optimal sparse singular value decomposition for high-dimensional high-order data. Journal of the American Statistical Association\/ 114\/ (528), 1708--1725

2019
[45]

Zhang, A. R. and D. Xia (2018). Tensor svd: Statistical and computational limits. IEEE Transactions on Information Theory\/ 64\/ (11), 7311--7338

2018
[46]

Wang, and L

Zhao, P., H. Wang, and L. Feng (2024). Spatial sign based principal component analysis for high dimensional data. arXiv preprint arXiv:2409.13267\/

2024 arXiv
[47]

Zhao, P. and B. Yu (2006). On model selection consistency of lasso. Journal of Machine learning research\/ 7\/ (Nov), 2541--2563

2006
[48]

Li, and H

Zhou, H., L. Li, and H. Zhu (2013). Tensor regression with applications in neuroimaging data analysis. Journal of the American Statistical Association\/ 108\/ (502), 540--552

2013
[49]

Zhou, J., W. W. Sun, J. Zhang, and L. Li (2023). Partially observed dynamic tensor response regression. Journal of the American Statistical Association\/ 118\/ (541), 424--439

2023
[50]

Zhou, S. (2014). Gemini: Graph estimation with matrix variate normal instances. The Annals of Statistics\/ 42\/ (2), 532--562

2014
[51]

Zou, C., L. Peng, L. Feng, and Z. Wang (2014). Multivariate sign-based high-dimensional tests for sphericity. Biometrika\/ 101\/ (1), 229--236

2014

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Arashi, M. (2023). Some theoretical results on the tensor elliptical distribution. Journal of Data Science and Modeling\/ 1\/ (2), 29--41

work page 2023

[2] [2]

Shibata, and M

Baba, K., R. Shibata, and M. Sibuya (2004). Partial correlation and conditional correlation as measures of conditional independence. Australian & New Zealand Journal of Statistics\/ 46\/ (4), 657--664

work page 2004

[3] [3]

Bickel, P. J. and E. Levina (2008). Covariance regularization by thresholding. The Annals of Statistics\/ 36\/ (6), 2577--2604

work page 2008

[4] [4]

Liu, and X

Cai, T., W. Liu, and X. Luo (2011). A constrained l1 minimization approach to sparse precision matrix estimation. Journal of the American Statistical Association\/ 106\/ (494), 594--607

work page 2011

[5] [5]

Cai, T. T., W. Liu, and H. H. Zhou (2016). Estimating sparse precision matrix: Optimal rates of convergence and adaptive estimation. The Annals of Statistics\/ 44\/ (2), 455--488

work page 2016

[6] [6]

High-Dimensional Tensor Classification with CP Low-Rank Discriminant Structure

Chen, E., Y. Han, and J. Li (2024). High-dimensional tensor classification with cp low-rank discriminant structure. arXiv preprint arXiv:2409.14397\/

work page Pith review arXiv 2024

[7] [7]

Chen, E. Y. and J. Fan (2023). Statistical inference for high-dimensional matrix-variate factor models. Journal of the American Statistical Association\/ 118\/ (542), 1038--1055

work page 2023

[8] [8]

Statistical Inference for Ultrahigh Dimensional Location Parameter Based on Spatial Median

Cheng, G., L. Peng, and C. Zou (2023). Statistical inference for ultrahigh dimensional location parameter based on spatial median. arXiv preprint arXiv:2301.03126\/

work page Pith review arXiv 2023