Pith. sign in

REVIEW 3 major objections 3 minor 33 references

SOFARI-R: High-Dimensional Manifold-Based Inference for Latent Responses

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Bias-corrected estimators give valid confidence intervals for latent right factor vectors in sparse SVD multi-response regression.

desk verdict Solid extension of SOFARI to right-factor inference, but the weakly orthogonal branch leans on an unproved imported lemma from the companion paper. read the letter →

arxiv 2504.17874 v1 pith:NWEKX5XQ submitted 2025-04-24 stat.ME cs.LGstat.ML

Zemin Zheng , Xin Zhou , Jinchi Lv This is my paper · ORCID
classification stat.MEcs.LGstat.ML MSC 62H2562F1262J05
keywords multi-tasklearningsparsesingularvaluedecompositionmanifold-basedinferenceStiefelmanifoldNeymanorthogonalitybiascorrectionasymptoticnormalityhigh-dimensionalregression
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

This paper tries to establish valid statistical inference—confidence intervals and tests—for the latent right factor vectors in the sparse singular value decomposition of a high-dimensional multi-response regression coefficient matrix. The difficulty is that, from the response side, signals from different SVD layers are correlated through the design matrix, so the usual manifold constraint on left singular vectors does not separate the nuisance layers. The authors couple each left singular vector with the design matrix, rescale the coupled object to unit length, and constrain it to a Stiefel manifold; a Neyman near-orthogonal score then produces bias-corrected estimators. The paper proves asymptotic normality for a strong-orthogonality variant and, using hard-thresholded debiased left-factor estimates with layer subtraction, for a weak-orthogonality variant, both with consistent variance estimates. If the theorems hold, any sparse linear combination of each latent right factor vector has a valid asymptotic normal pivot, making response selection with uncertainty quantification possible in multi-task learning and including high-dimensional PCA as a special case.

What carries the argument

The load-bearing construction is the rescaled coupled left factor $u_i=(l_i^{\top}\hat{\Sigma}l_i)^{-1/2}n^{-1/2}Xl_i$, paired with $v_i=(l_i^{\top}\hat{\Sigma}l_i)^{1/2}d_i r_i$; requiring $u_i^{\top}u_i=1$ puts these $u_i$ on Stiefel manifolds—sets of unit-length vectors—even though the original left singular vectors are not orthogonal after the design map. The constrained least-squares loss $L(v_k,\eta_k)=(2n)^{-1}\|Y-\sum_i\sqrt{n}\,u_i v_i^{\top}\|_F^2$ with $u_i^{\top}u_i=1$ and $v_i^{\top}v_j=0$ generates a modified score $\tilde{\psi}_k=\partial L/\partial v_k-M^{(k)}\partial L/\partial\eta_k$; Propositions 6 and 7 give the matrices $M^{(k)}$ and $W_k$ so that the score is Neyman near-orthogonal on the manifold and the bias term in the debiased estimate vanishes. For weakly orthogonal factors, the loss subtracts the other layers using hard-thresholded debiased left-factor estimates $\hat{\mu}_i^t$, and Propositions 3 and 4 give $M_k=-(\hat{v}_k^{\top}\hat{v}_k)^{-1}\hat{v}_k\hat{u}_k^{\top}$ and a closed-form $W_k$; the approximation errors from this substitution are absorbed into the asymptotic expansion rather than ignored.

What would settle it

Run the weakly orthogonal SOFARI-R on a design where the cross-layer correlations $|l_j^{\top}\hat{\Sigma}l_k|$ decay exactly like $c n^{-1/2}$ (so weak orthogonality fails at the boundary) and where singular-value gaps are only as large as Condition 3 requires; if the coverage of nominal 95% intervals over many replications falls well below 0.93, the $o_p(1)$ remainder in Theorem 3 is not what the proof claims. A second check is to test Condition 6 directly in the paper's own simulations: if $\min_{j\in\operatorname{supp}(\mu_i^*)}|\mu_{ij}^*|\ge C n^{-\alpha}$ with $\alpha<1/2$ does not hold, the hard-thresholding lemma cannot justify the support recovery.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that for each layer $k$, the debiased estimator $\hat{v}_k$ satisfies $\sqrt{n}\,a^{\top}(\hat{v}_k-v_k^*)=h_k+t_k$ for any direction $a$ with $\|a\|_0\le m$, where the leading term $h_k$ is asymptotically $N(0,\nu_k^2)$ with explicit variance $\nu_k^2$ and the remainder satisfies $t_k=o_p(1)$ under the stated conditions. The strong-orthogonality variant attains this with a simpler variance and no precision-matrix estimation; the weak-orthogonality variant attains it with a variance that also includes the debiased left-factor estimates of the other layers. Theorems 2 and 4 provide plug-in variance estimates whose error is of the same order as the SOFAR estimation error, so the pivots can be used to build confidence intervals for every component of every significant right factor vector. This completes inference on all SVD components of the coefficient matrix, after earlier work covered the left factor vectors and singular values.

Load-bearing premise

The load-bearing premise is that the companion paper's debiased left-factor estimators $\hat{\mu}_i$ are asymptotically normal under the stated conditions (imported as Lemma 10), and that the hard-thresholding signal-separation Condition 6 holds, since Theorem 3's expansion collapses if either fails.

Editorial extensions

If this is right

  • For any layer and any sparse direction, the debiased estimator provides an asymptotically normal pivot for the corresponding linear combination of the latent right factor vector, so componentwise confidence intervals reach the nominal coverage as the sample size grows.
  • The strong-orthogonality variant needs no approximate inverse of the design Gram matrix and imposes no sparsity condition on the precision matrix, widening the settings where such inference is available.
  • The weak-orthogonality variant extends the method to correlated latent factors, provided the debiased left-factor distribution theory and the hard-thresholding signal-separation condition hold.
  • Together with inference for left factor vectors and singular values, the results complete manifold-based inference for all sparse SVD components of the coefficient matrix in multi-response regression.
  • When the design matrix is the identity, the right factor vectors are principal components, so the strong-orthogonality variant gives asymptotic distributions for all significant principal-component directions, not just the first eigenvector.

Reading between the lines

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

  • The per-layer expansions suggest joint confidence regions across several right factor vectors could be built from the same normal terms, but the paper does not develop simultaneous inference.
  • Because the strong-orthogonality variant avoids precision-matrix estimation, the coupling-rescaling construction may transfer to designs where the Gram matrix is singular or the precision matrix is not sparse.
  • The hard-thresholding step offers a diagnostic: if the support of the thresholded left-factor estimate changes when the threshold is moved around $\log n/\sqrt{n}$, Condition 6 may be violated and weak-orthogonality intervals deserve caution.
  • Theorem 3 inherits any unproved gap in the companion lemma on debiased left-factor normality; a self-contained proof of that lemma inside this framework would remove the main external dependency.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper develops SOFARI-R, a debiased-inference framework for the latent right factor vectors v*_k in the sparse SVD of a high-dimensional regression coefficient matrix C* = sum_i d*_i l*_i r*_i^T. Two variants are introduced: SOFARI-Rs for strongly orthogonal factors, which couples left singular vectors with the design matrix and rescales them to build Stiefel-manifold score functions, and SOFARI-R for weakly orthogonal factors, which removes other layers using hard-thresholded SOFARI debiased left-factor estimates. Theorems 1-2 establish asymptotic normality and variance estimation for the strong case; Theorems 3-4 do the same for the weak case. The paper reports simulation studies with coverage close to nominal and a FRED-MD forecasting application.

Significance. The paper addresses a genuinely asymmetric inference problem: right singular vectors cannot be handled by transposing the SOFARI construction, because layer signals are correlated through the design matrix. The coupling-and-rescaling idea for strong orthogonality is novel and avoids inverse Hessian or precision-matrix estimation in that variant. If the weak-orthogonality theorem is valid, the method provides distributional inference for all significant right factors in this model class. The supplement gives detailed proofs and lemmas for the strong case and for most of the weak-case remainder analysis. However, the weak-case theory depends on an unproved imported lemma from a companion preprint, and Theorem 3 lacks explicit rate conditions that make its stated error term o_p(1); these issues need resolution before the central claims can be accepted.

major comments (3)
  1. [Section 3.3 and Supplement A.3 (Theorem 3)] Theorem 3 states that t_k = O_p( m^{1/2} d*_1 d*^{-2}_{r*} kappa_n + s_u^{1/2} r* (d*_1 d*^{-2}_{r*} gamma_n + kappa'_n max{1, d*_{r*}^{-1}, d*_{r*}^{-2}}) ), but it does not state conditions under which this bound is o_p(1). Asymptotic normality requires t_k = o_p(1); without such conditions the displayed result is only a rate bound for an error term of unknown size. In addition, the proof of Lemma 9 in Supplement C.9 applies Lemma 10 to the sparse vector (Sigma_hat mu*_k)_{t_i}, whose l0-norm is at most s_u. That application requires s_u^{1/2} kappa_n^{(i)} = o(1), a condition that is not stated in Theorem 3 and is not shown to follow from Conditions 1-3, 5-6. Please add explicit rate conditions and verify that they hold in the simulation settings.
  2. [Supplement B, Lemma 10 (and its use in Lemma 9)] The weak-orthogonality branch is built on Lemma 10, quoted verbatim as Theorem 3 of the authors' companion preprint [24]. This imported lemma supplies the asymptotic normality and covariance of the debiased left-factor estimators, which enter the distribution term h_{ui} in Theorem 3, the variance formula (A.69), the support-recovery Lemma 15, and the error bound in Theorem 4. The present manuscript neither proves Lemma 10 nor verifies that the companion paper's conditions match this setup. Since [24] is an arXiv preprint and not a peer-reviewed reference at the time of submission, the weak-orthogonality results are conditional on an unverified imported result. The authors should either provide a self-contained proof of Lemma 10 in the supplement or rely on a published version with explicitly checked conditions.
  3. [Condition 6 and Section 5 (FRED-MD application)] Lemma 15's support recovery and the hard-thresholding step rely on Condition 6, which requires signal strengths at least C_u n^{-alpha} with alpha < 1/2 and negligible total off-support mass. The paper does not discuss whether this condition is plausible for the FRED-MD design, nor whether the threshold log n / sqrt(n) is appropriate for n = 654. The confidence sets for the weakly orthogonal variant in Section 5 therefore depend on an unexamined signal-separation assumption. At minimum, the authors should report a diagnostic or state explicitly how many components are above the threshold and discuss the plausibility of Condition 6 for the economic data.
minor comments (3)
  1. [Supplement C.8, Lemma 8] The notation d*_{k*} appears in the statement of Lemma 8 and in its proof, but d*_{k*} is not defined. It should be d*_{r*} (or d*_k) consistently.
  2. [Supplement B, Lemma 10] In the variance formula of Lemma 10, the symbols z*_{kk} and Sigma_hat*_k are undefined. They should be mu*_k^T Sigma_hat mu*_k and Sigma_hat mu*_k r*_k^T, respectively, matching the corresponding expression in Lemma 9.
  3. [Section 1 and Table 1] The introduction and Table 1 state that SOFARI-R does not require an approximate inverse of Sigma_hat, but Definition 2 and Section 2.2 show that the weakly orthogonal SOFARI-R does use Sigma_hat via the debiased left-factor estimators. Please clarify that this advantage holds only for the strongly orthogonal SOFARI-Rs variant.

Circularity Check

1 steps flagged · score 4.0 of 10

Weak-orthogonality Theorem 3 inherits its distribution term from unproved Lemma 10, quoted from the same authors' companion preprint; strong-orthogonality branch remains self-contained.

  1. self citation load bearing [Supplementary Material, Section B, Lemma 10; used in Lemma 9 and Theorem 3]
    "Lemma 10 (Theorem 3, Zheng et al. [24]). Assume that Conditions 1–3 and 5 hold, and eC, bΘ satisfy Definitions 1 and 2, respectively. Then for each given k with 1≤ k≤ r∗ and an arbitrary vector a∈A ={a∈ Rp :∥a∥0≤m,∥a∥2 = 1} satisfying m1/2κ(k) n =o(1), we have that √naT (bµk−µ∗ k) =huk +tuk,"

    The paper's Theorem 3 for weakly orthogonal factors decomposes √n a^T(bv_k − v*_k) and, in Lemma 9, converts the hard-thresholded left-factor term into ∑_{i≠k} ω_{k,i} h_i((Σ̂ μ*_k)_{t_i}) plus a remainder. The h_i terms are exactly the normal variates supplied by Lemma 10, quoted verbatim from the same authors' companion preprint [24] and not proved in this manuscript; the covariance formula (A.69) and Theorem 4's variance estimator inherit W*_{ui} and M*_{ui} from that lemma. Lemma 15's support recovery also uses the root-n normality of bµ from Lemma 10. Thus the weak-orthogonality branch of the central claim reduces to an imported, unverified theorem by the same authors rather than to a proof within this paper.

full rationale

The strong-orthogonality branch (Theorems 1 and 2) is self-contained: it uses Propositions 5–7 and Lemmas 1–4, all proved in the supplement, with no fitted input renamed as a prediction. The method is a debiasing construction against the SOFAR estimator, not a fit-to-target, so there is no self-definitional or fitted-input circularity. The weak-orthogonality branch is the concern: Theorem 3 and Theorem 4 depend on Lemma 9, whose distribution term is the asymptotic normality of the debiased left-factor estimators bµ_i stated as Lemma 10 from Zheng et al. [24], by the same authors, and not proved here. This is not circular in the strict 'prediction equals input by construction' sense, because the paper does not fit the right-factor distribution to data; it is a load-bearing self-citation. If Lemma 10 is accepted as an established theorem, the derivation chain is valid. But as presented, the central weak-orthogonality claim is conditional on an unverified imported result, so the score is 4 rather than 0; it is not higher because the strong-orthogonality claim is independent and no fitted-input/prediction equivalence is present.

Assumptions & free parameters 3 free parameters · 8 assumptions · 0 invented entities

The method rests on six regularity conditions (Conditions 1-6) and on the SOFAR estimator meeting the error bounds in Definition 1 (established in [17]); the weak-orthogonality variant additionally imports the debiased left-factor normality from the companion SOFARI paper [24] (Lemma 10) and requires an approximate inverse of Sigma_hat (Definition 2). No constants are fitted to make the theorems true; the rank and penalty parameters are practical tuning inputs.

free parameters (3)
  • Rank r* = estimated via self-tuning of [2]; r=3 in simulations and application
    The number of SVD layers is an input to every construction (M(k), W_k, summations over i != k); theory assumes a consistent estimate, simulations and FRED-MD application set it to 3.
  • Hard-thresholding level = logn/sqrt(n) (fixed rate)
    Used to sparsify debiased left estimates in SOFARI-R; the constant is rate-based, not tuned to data.
  • SOFAR-L penalty parameter = not specified (inherited from SOFAR [17])
    Initial estimator C_hat in Definition 1 depends on a penalty tuning that is not detailed; simulations likely use cross-validation.
assumptions (8)
  • domain assumption Gaussian error E ~ N(0, I_n tensor Sigma_e) with bounded spectrum (Condition 1)
    Used throughout for concentration of n^{-1}X^TE and normal distribution of h_k; authors note relaxation to non-Gaussian possible but not proved here.
  • domain assumption Sparse eigenvalue bounds on Sigma_hat (Condition 2) and eigengap condition d*_i^2 - d*_{i+1}^2 >= delta_1 d*_i^2 (Condition 3)
    Needed for identifiability of layers and invertibility of W_k (Lemma 12).
  • domain assumption Strong orthogonality sum_{j != k} |l*_j^T Sigma_hat l*_k| = o(n^{-1/2}) (Condition 4) or weak orthogonality sum_{j >= k+1} (d*_j^2/d*_k)|l*_j^T Sigma_hat l*_k| = o(n^{-1/2}) (Condition 5)
    Defines the two regimes covered; Condition 4 is used in Lemmas 1-4; Condition 5 in Lemmas 5-9.
  • domain assumption Acceptable SOFAR estimator C_hat with error bounds of Definition 1 (established by Theorem 2 of [17])
    All proofs condition on this event E0; the rates gamma_n and eta_n enter every error term.
  • domain assumption Approximate inverse Theta_hat of Sigma_hat exists with row-sparsity s_max and entrywise error (Definition 2), needed for the SOFARI debiased left estimators in the weak case
    The weakly orthogonal variant uses the SOFARI mu_hat_i from [24], which requires Theta_hat; not needed for strong case.
  • ad hoc to paper Asymptotic normality and variance formula for SOFARI left debiased estimator mu_hat_i (Lemma 10, Theorem 3 of [24])
    Imported unproved from the authors' companion preprint; used in Lemma 9 and the distribution of SOFARI-R.
  • domain assumption Support recovery of hard-thresholded debiased estimates supp(mu_hat^t_i)=supp(mu*_i) (Lemma 15) under Condition 6 signal strength
    Requires nonzero signals >= C n^{-alpha}, alpha < 1/2; not checked in real data.
  • standard math Standard linear algebra results (Sherman-Morrison-Woodbury, Levy-Desplanques, Stiefel manifold Taylor expansion)
    Used in Propositions 4, 6, 7 and Lemma 12; accepted background.

how reviews work

0 comments
Cite this review

Pith. "Pith review of SOFARI-R: High-Dimensional Manifold-Based Inference for Latent Responses." pith.science (2026). https://pith.science/paper/NWEKX5XQ

@misc{pith2026250417874,
  author       = {Pith},
  title        = {Pith review of: SOFARI-R: High-Dimensional Manifold-Based Inference for Latent Responses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NWEKX5XQ}},
  note         = {Machine review of arXiv:2504.17874}
}
read the original abstract

Data reduction with uncertainty quantification plays a key role in various multi-task learning applications, where large numbers of responses and features are present. To this end, a general framework of high-dimensional manifold-based SOFAR inference (SOFARI) was introduced recently in Zheng, Zhou, Fan and Lv (2024) for interpretable multi-task learning inference focusing on the left factor vectors and singular values exploiting the latent singular value decomposition (SVD) structure. Yet, designing a valid inference procedure on the latent right factor vectors is not straightforward from that of the left ones and can be even more challenging due to asymmetry of left and right singular vectors in the response matrix. To tackle these issues, in this paper we suggest a new method of high-dimensional manifold-based SOFAR inference for latent responses (SOFARI-R), where two variants of SOFARI-R are introduced. The first variant deals with strongly orthogonal factors by coupling left singular vectors with the design matrix and then appropriately rescaling them to generate new Stiefel manifolds. The second variant handles the more general weakly orthogonal factors by employing the hard-thresholded SOFARI estimates and delicately incorporating approximation errors into the distribution. Both variants produce bias-corrected estimators for the latent right factor vectors that enjoy asymptotically normal distributions with justified asymptotic variance estimates. We demonstrate the effectiveness of the newly suggested method using extensive simulation studies and an economic application.

Figures

Figures reproduced from arXiv: 2504.17874 by the authors.

Figure 1
Figure 1. The kernel density estimates (KDEs) for the distributions of SOFARI-R estimates on the latent right factor vectors in different sparse SVD layers against the target standard normal density based on 1000 replications for setting 1 in Section 4. Left panel: the KDEs of T1,1, T2,4, and T3,7; right panel: the KDEs of T1,15, T2,15, and T3,15, all viewed from top to bottom. The blue curves represent the KDEs for SOFARI-R … view at source ↗
Figure 2
Figure 2. Bar charts of the significant components in the top three latent right factor vec [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 32 canonical work pages

  1. [24]

    Zheng, Z., X. Zhou, Y. Fan, and J. Lv (2024). SOFARI: high-dimensional manifold- based inference. arXiv preprint arXiv:2309.15032

    arXiv 2024

  2. [1]

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

    work page 2008

  3. [2]

    Bing, X. and M. H. Wegkamp (2019). Adaptive estimation of the rank of the coeffi- cient matrix in high-dimensional multivariate response regression models. The Annals of Statistics 47 (6), 3157–3184. 17

    work page 2019

  4. [3]

    She, and H

    Bunea, F., Y. She, and H. M. Wegkamp (2012). Joint variable and rank selection for parsimonious estimation of high-dimensional matrices. The Annals of Statistics 40 , 2359– 2388

    work page 2012

  5. [4]

    Cai, T. and W. Liu (2011). Adaptive thresholding for sparse covariance matrix estimation. Journal of the American Statistical Association 106 (494), 672–684

    work page 2011

  6. [5]

    Chan, and N

    Chen, K., K.-S. Chan, and N. C. Stenseth (2012). Reduced rank stochastic regression with a sparse singular value decomposition. Journal of the Royal Statistical Society Series B 74 (2), 203–221

    work page 2012

  7. [6]

    Dong, and K.-S

    Chen, K., H. Dong, and K.-S. Chan (2013). Reduced rank regression via adaptive nuclear norm penalization. Biometrika 100 , 901–920

    work page 2013

  8. [7]

    Chen, K., R. Dong, W. Xu, and Z. Zheng (2022). Fast stagewise sparse factor regression. Journal of Machine Learning Research 23 (271), 1–45

    work page 2022

Show all 33 references
  1. [8]

    Chen, L. and J. Z. Huang (2012). Sparse reduced-rank regression for simultaneous di- mension reduction and variable selection. Journal of the American Statistical Associa- tion 107 (500), 1533–1545

    2012

  2. [9]

    Chetverikov, M

    Chernozhukov, V., D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey, and J. Robins (2018). Double/debiased machine learning for treatment and structural param- eters. The Econometrics Journal 21 (1), 1–68

    2018

  3. [10]

    Fan, C.-K

    Chi, C.-M., Y. Fan, C.-K. Ing, and J. Lv (2025). High-dimensional knockoffs inference for time series data. Journal of the American Statistical Association, to appear

    2025

  4. [11]

    Horn, R. A. and C. R. Johnson (2012). Matrix Analysis. Cambridge University Press

    2012

  5. [12]

    Jankov´ a, J. and S. van de Geer (2021). De-biased sparse PCA: inference for eigen- structure of large covariance matrices. IEEE Transactions on Information Theory 67 (4), 2507–2527

    2021

  6. [13]

    McCracken, M. W. and S. Ng (2016). FRED-MD: a monthly database for macroeco- nomic research. Journal of Business & Economic Statistics 34 (4), 574–589

    2016

  7. [14]

    Meinshausen, N. and P. B¨ uhlmann (2006). High-dimensional graphs and variable selec- tion with the lasso. The Annals of Statistics 34 (3), 1436–1462

    2006

  8. [15]

    Mishra, A., D. K. Dey, and K. Chen (2017). Sequential co-sparse factor regression. Journal of Computational and Graphical Statistics 26 (4), 814–825

    2017

  9. [16]

    Neyman, J. (1959). Optimal asymptotic tests of composite statistical hypotheses. In U. Grenander (Ed.), Probability and Statistics , 213–234

    1959

  10. [17]

    Uematsu, Y., Y. Fan, K. Chen, J. Lv, and W. Lin (2019). SOFAR: large-scale association network learning. IEEE Transactions on Information Theory 65 (8), 4924–4939. 18

    2019

  11. [18]

    B¨ uhlmann, Y

    van de Geer, S., P. B¨ uhlmann, Y. Ritov, and R. Dezeure (2014). On asymptotically optimal confidence regions and tests for high-dimensional models. The Annals of Statis- tics 42 (3), 1166–1202

    2014

  12. [19]

    Varah, J. M. (1975). A lower bound for the smallest singular value of a matrix. Linear Algebra and Its Applications 11 (1), 3–5

    1975

  13. [20]

    Wang, W. and J. Fan (2017). Asymptotics of empirical eigenstructure for high dimen- sional spiked covariance. The Annals of Statistics 45 (3), 1342

    2017

  14. [21]

    Chen, and J

    Yan, Y., Y. Chen, and J. Fan (2024). Inference for heteroskedastic pca with missing data. The Annals of Statistics 52 (2), 729 – 756

    2024

  15. [22]

    Zheng, Z., M. T. Bahadori, Y. Liu, and J. Lv (2019). Scalable interpretable multi- response regression via SEED. Journal of Machine Learning Research 20 (107), 1–34

    2019

  16. [23]

    Lv, and W

    Zheng, Z., J. Lv, and W. Lin (2021). Nonsparse learning with latent variables. Opera- tions Research 69 (1), 346–359

    2021

  17. [25]

    SOF ARI-R: High-Dimensional Manifold-Based Inference for Latent Responses

    Zou, C., Y. Ke, and W. Zhang (2022). Estimation of low rank high-dimensional multi- variate linear models for multi-response data. Journal of the American Statistical Associ- ation 117 (538), 693–703. 19 Supplementary Material to “SOF ARI-R: High-Dimensional Manifold-Based Inf...

    2022

  18. [26]

    Hence, combining the above terms leads to A12≤cγnd∗ 1d∗−2 k

    (A.29) Then we can deduce that |aTWkMkMT k WT kea−aTW∗ kM∗ kM∗T k W∗T k a| ≤|aTWkMk(MT k WT ka− M∗T k W∗T k a)| +|(aTWkMk−aTW∗ kM∗ k)M∗T k W∗T k a| ≤∥aTWk∥2∥Mk∥2∥MT k WT ka− M∗T k W∗T k a∥2 +∥aTWkMk−aTW∗ kM∗ k∥2∥M∗T k ∥2∥W∗T k a∥2 ≤cγnd∗−4 k d∗ 1. Hence, combining the above te...

  19. [27]

    it can be seen that |aTWkeruk|≤∥ aTWk∥2∥eruk∥2≤c(r∗ +su +sv)η4 n{n−1 log(pq)}. 44 Therefore, combining the above terms gives that |aTWk(eψk(evk,eηk)− eψk(evk,η∗ k))| ≤cm1/2 max{(r∗ +su +sv)1/2,η 2 n}(r∗ +su +sv)η2 n{n−1 log(pq)}d∗−1 k , which completes the proof of Lemma 7. C....

  20. [28]

    From (A.111), we have|l∗T i bΣl∗ i|−1 = 54 |µ∗T i bΣµ∗ i|−1d∗2 i ≤c

    (A.116) Observe thatL∗T 0 L∗ 0 = diag{(l∗T 1 bΣl∗ 1)−1,··· , (l∗T r∗ bΣl∗ r∗)−1}. From (A.111), we have|l∗T i bΣl∗ i|−1 = 54 |µ∗T i bΣµ∗ i|−1d∗2 i ≤c. Then it holds that sup bTb=1 ∥L∗ 0b∥2 2 = sup bTb=1 bTL∗T 0 L∗ 0b = sup bTb=1 bT diag{(l∗T 1 bΣl∗ 1)−1,··· , (l∗T r∗ bΣl∗ r∗)−...

  21. [29]

    By (A.111), we have ∥eDl∥2≤cd∗−1 r∗ . Furthermore, it holds that ∥eL0− L∗ 0∥0 =∥eUdeDl− U∗ dD∗ l∥0 ≤∥ (eUd− U∗ d)eDl∥0 +∥U∗ d(eDl− D∗ l )∥0 ≤∥ eUd− U∗ d∥0 +∥U∗ d∥0 ≤ 3(r∗ +su +sv), where the last step above holds due to Definition 1. Based on the sparsity of eL0− L∗ 0, similar...

  22. [30]

    (A.122) Next we analyze matrix V following similar analysis as for U

    Hence, for sufficiently large n, we have that ∥eL0∥2≤∥ L∗ 0∥2 +∥eL0− L∗ 0∥2≤c, ∥eU∥2≤∥ U∗∥2 +∥eU− U∗∥2≤c. (A.122) Next we analyze matrix V following similar analysis as for U. Note that V∗ = (v∗ 1,··· ,v∗ r∗) = (µ∗T 1 bΣµ∗ 1)1/2r∗ 1,··· , (µ∗T r∗ bΣµ∗ r∗)1/2r∗ r∗ . Similar to ...

  23. [31]

    Similarly, we have that ∥eV∥2≤cd∗

  24. [32]

    Then we see that V∗ = R∗D∗ v and eV = eReDv

    Further, let us define R∗ = (r∗ 1,··· ,r∗ r∗), D∗ v = diag{(µ∗T 1 bΣµ∗ 1)1/2,··· , (µ∗T r∗ bΣµ∗ r∗)1/2}, and eRd,eDv analogously. Then we see that V∗ = R∗D∗ v and eV = eReDv. In view of (A.109) and (A.113), it holds that ∥eDv∥2≤cd∗ 1, ∥eDv− D∗ v∥2≤cγn. By definition, we have∥R...

  25. [33]

    Finally, for term ∥aTWk∥2, it follows from the triangle inequality that for sufficiently 64 large n, ∥aTWk∥2≤∥aT (Wk− W∗ k)∥2 +∥aTW∗ k∥2≤c

    (A.136) Combining (A.135) and (A.136) yields that ∥aT (Wk− W∗ k)∥2≤cγnd∗ 1d∗−2 k . Finally, for term ∥aTWk∥2, it follows from the triangle inequality that for sufficiently 64 large n, ∥aTWk∥2≤∥aT (Wk− W∗ k)∥2 +∥aTW∗ k∥2≤c. This completes the proof of Lemma 14. C.14 Lemma 15 an...

    2002

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.