High-Dimensional Regularized Additive Matrix Autoregressive Model

Debika Ghosh; Nilanjana Chakraborty; Samrat Roy

REVIEW 3 major objections 4 minor 36 references

High-Dimensional Regularized Additive Matrix Autoregressive Model

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read An additive row-plus-column matrix autoregressive model is fitted with convex penalties and shown to have finite-sample error bounds whose dominant terms match sparse-VAR and matrix-decomposition rates.

desk verdict Sensible extension of additive MAR, but the missing gauge condition makes the central error bound not well-posed; worth referee time, not acceptance. read the letter →

arxiv 2506.01403 v1 pith:CYPFAXVM submitted 2025-06-02 stat.ME

Debika Ghosh , Samrat Roy , Nilanjana Chakraborty This is my paper

classification stat.ME MSC 62M1062H1262J07

keywords matrixautoregressivehigh-dimensionaltimeserieslow-rankplussparsedecompositionnuclearnormregularizationalternatingblockminimizationidentifiabilityconvexoptimization

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

The paper proposes replacing the bilinear matrix autoregressive law with the additive law $Y_t = A Y_{t-1} + Y_{t-1} B'$ so that row-wise and column-wise temporal dependence can be estimated and interpreted separately. It argues that this additive form stays convex when the transition matrices are penalized, and that it can carry low-rank, sparse, or low-rank-plus-sparse transition structure. The central theoretical claim is a finite-sample error bound: with high probability, the squared Frobenius error in estimating all four components is controlled by sparsity terms of order $s_1 Q_1^2 \log(d_1)/T$ and $s_2 Q_2^2 \log(d_2)/T$ plus rank terms of order $R_1 Q_1^2 d_1/T$ and $R_2 Q_2^2 d_2/T$. If this bound is correct, the model recovers the transition structure of matrix time series at near-optimal high-dimensional rates while keeping the optimization problem convex.

What carries the argument

The machinery is the additive transition structure $Y_t = (L_1+S_1)Y_{t-1} + Y_{t-1}(L_2+S_2)'$ with nuclear-norm penalties on $L_1, L_2$ and $\ell_1$ penalties on $S_1, S_2$, minimized by alternating block minimization in which each block update is a convex penalized regression. The theoretical engine is the decomposability of the $\ell_1$ and nuclear-norm regularizers: Lemmas 3.1 and 3.2 restrict the estimation error to a cone where the undesired part of the error is bounded by a constant multiple of the desired part, Assumption 3.1 supplies restricted strong convexity, and Assumption 3.2 (the spikiness bounds $\|L_1\|_\infty \le \alpha_1/\sqrt{d_1^2}$, $\|L_2\|_\infty \le \alpha_2/\sqrt{d_2^2}$) separates the low-rank pieces from the sparse pieces. The identity carrying the argument is the basic inequality relating the empirical loss difference to the regularizer gap, combined with the cone constraint $C(\hat\Delta^B) \le 3C(\hat\Delta^A)$.

What would settle it

Simulate $Y_t = A Y_{t-1} + Y_{t-1}B' + E_t$ once with the pair $(A,B)$ and once with $(A+cI, B-cI)$ using the same noise realization; because the two series are identical, any procedure that returns different estimates for the two runs, or any error bound that claims to recover $A$ and $B$ separately, can be checked directly by comparing the fitted pairs with the two generating pairs.

Watch

Extended reading notes

Core claim

The core discovery is that the additive matrix autoregressive model $Y_t = (L_1+S_1)Y_{t-1} + Y_{t-1}(L_2+S_2)' + E_t$ can be estimated by minimizing a jointly convex objective with nuclear-norm penalties on $L_1, L_2$ and $\ell_1$ penalties on $S_1, S_2$, and that the resulting estimator obeys the non-asymptotic error bound of Theorem 3.1. Under Gaussian noise with covariance $\Sigma_1 \otimes I_{d_2} + I_{d_1} \otimes \Sigma_2$ and a spikiness condition on the low-rank components, the sum of squared Frobenius errors over $\hat L_1, \hat L_2, \hat S_1, \hat S_2$ is bounded, with probability $1 - \max(e^{-c_1 \log d_1}, e^{-c_2 \log d_2})$, by $s_1\{c_1 Q_1^2 \frac{2\log d_1}{T} + \frac{c_2\gamma^2\alpha_1^2}{d_1^2}\} + s_2\{c_3 Q_2^2 \frac{2\log d_2}{T} + \frac{c_4\gamma^2\alpha_2^2}{d_2^2}\} + \frac{c_5 Q_1^2 R_1 \, 2d_1}{T} + \frac{c_6 Q_2^2 R_2 \, 2d_2}{T}$. The additive structure lets low-rank parts represent shared or baseline dynamics and sparse parts represent idiosyncratic links, and the paper demonstrates better rolling-window forecast accuracy than the bilinear matrix autoregressive model and a vectorized sparse VAR on Eurozone macroeconomic data.

Load-bearing premise

The whole estimation target presupposes that row-wise and column-wise transition matrices can be told apart, but the additive model is unchanged if $A$ is replaced by $A+cI$ and $B$ by $B-cI$ for any scalar $c$; the paper's spikiness condition separates low-rank from sparse parts, not this row-versus-column shift, so the split is identified only after an extra constraint that the paper does not state.

Editorial extensions

If this is right

Row-wise and column-wise transition matrices can be estimated under high-dimensional scaling without solving a non-convex problem, because each alternating block update is a convex nuclear-norm or $\ell_1$-penalized regression.
The error bound makes the sample-size requirement explicit: roughly $T \gtrsim s \log d$ for the sparse components and $T \gtrsim R d$ for the low-rank components, matching rates known from sparse vector autoregression and matrix decomposition.
The estimated low-rank and sparse pieces acquire a substantive reading: low-rank parts are shared or baseline temporal dependencies, while sparse parts are idiosyncratic links, which is exactly the interpretation used in the Eurozone application.
On the real dataset, the additive model produces lower forecast RMSE than the bilinear MAR and vectorized sparse VAR baselines at horizons 1, 2, and 3, suggesting that the additive form is not merely interpretable but also predictively competitive.

Reading between the lines

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

The row-versus-column shift ambiguity is left implicit by the paper: since $(A+cI, B-cI)$ generates the same series as $(A,B)$, the split between indicator effects and country effects is meaningful only after choosing a gauge, such as centering one transition matrix, which the paper does not impose.
The spikiness condition separates the low-rank part from the sparse part within each transition matrix, but it does not resolve the ambiguity between $A$ and $B$ themselves; the real-data network interpretation therefore depends on an unstated identifiability choice.
A direct testable extension would be to fit the model to data generated from $(A,B)$ and again from $(A+cI, B-cI)$ with the same noise: the two data sets are identical, so the fitted transition pairs should reveal the gauge dependence directly, and any error bound that claims separate recovery of $A$ and $B$ would have to be qualified.

Share X Bluesky LinkedIn Reddit HN

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

Here's the short version: this is a sensible attempt to make additive matrix autoregression high-dimensional by adding low-rank-plus-sparse penalties, and the optimization is clean. But the main theorem is not well-posed. The map (A,B) -> (A+cI, B-cI) leaves the model unchanged, so the row-wise/column-wise split is unidentifiable, and nothing in the paper's assumptions fixes that. The proof structure is also shaky. So the theory, as written, does not establish what it claims.

What is genuinely new: the additive form is from Zhang (2024), and the contribution here is the regularized convex estimator and its finite-sample bound. That is a real extension, and the bound has the expected shape (s log d/T plus R d/T terms). The Eurozone application in Section 5 is a nice illustration of the interpretability the authors want.

The soft spots, in order of severity. First, the identifiability problem is load-bearing. For any c, (A+cI, B-cI) gives identical conditional means. The nuclear/l1 penalties will break the tie, but in a data-dependent way, and Assumption 3.2 (spikiness) only separates low-rank from sparse within each matrix, not the shift between A and B. So the true (L1,S1,L2,S2) that Theorem 3.1 bounds is an arbitrary member of an equivalence class, and the real-data interpretations in Section 5 are not identified. Second, the appendix proof labeled 'Lemma 3.1' actually uses Assumption 3.1 and derives the bound of Lemma 3.3; the stated decomposition inequalities (6)-(7) are never proved, and Lemma 3.2 is stated without proof. Third, the synthetic experiments choose tuning parameters with oracle knowledge of true ranks and supports, and the paper never benchmarks against Zhang's additive MAR, which is the natural competitor.

Overall: the idea is good and the paper is readable, but the theoretical core is not validated. I would send it to referees—it deserves a serious look—but I would reject in its current form, with the clear suggestion to add a gauge condition (trace or diagonal constraint), fix the proofs, and rerun the simulations without oracle tuning. If that revision comes back, it could be a solid contribution.

Referee Report

3 major / 4 minor

Summary. The paper proposes a regularized additive matrix autoregressive model, Y_t = (L1+S1) Y_{t-1} + Y_{t-1} (L2+S2)' + E_t, in which row-wise and column-wise temporal effects enter additively rather than multiplicatively. The transition matrices are estimated by minimizing a jointly convex objective that combines squared prediction error with nuclear-norm and ℓ1 penalties, using an alternating block minimization algorithm. The paper claims to address identifiability of the low-rank plus sparse components and derives a finite-sample error bound (Theorem 3.1) under Gaussian errors, with rates of order s1 log d1/T, s2 log d2/T, R1 d1/T, and R2 d2/T. The methodology is illustrated on synthetic data and on a quarterly Eurozone macroeconomic dataset.

Significance. If the main theorem were valid, the paper would offer a computationally attractive convex estimator for an interpretable additive matrix autoregression, with error rates matching standard high-dimensional sparse and low-rank benchmarks. The additive formulation itself is a sensible and potentially useful modeling choice, and the use of convex penalties is a practical strength. However, the central recovery claim is not well-posed because of an exact gauge invariance in the model, and the theoretical chain contains unproved lemmas. As it stands, the paper does not establish its advertised finite-sample guarantee, and the reported low-rank/sparse structure in the application is subject to an unresolved identifiability ambiguity.

major comments (3)

[Section 2.2, Eq. (1)] The conditional mean is invariant under (A,B) ↦ (A + c I_{d1}, B - c I_{d2}) for any scalar c, since c Y_{t-1} cancels with -c Y_{t-1}. The likelihood part of (3) is therefore invariant as well, and the data cannot distinguish among these parameter pairs. Assumption 3.2 (Eq. (10)) only controls the infinity norms of L1 and L2, which separates low-rank from sparse components within A and within B; it imposes no restriction on the cross-shift between A and B. The promises of identifiability in the abstract and Section 2.3 are not fulfilled by any condition in the paper. Consequently, the error e2 in (4) is measured against an arbitrary member of an equivalence class, and the estimator may converge to a different member; for instance, with d1 < d2 the pair (0, I_{d2}) is observationally equivalent to (I_{d1}, 0), and the penalty in (3) can strictly favor the latter when λS1 d1 < λS2 d2, regardless of T. Theorem 3.1 is therefore not a recovery statement for a uniquely defined target, and the Section 5 interpretation of the estimated low-rank and sparse components is not identifiable without a gauge condition such as trace(A)=0.
[Lemma 3.2] Lemma 3.2 is stated without proof and is load-bearing. The constraints (8) characterize the restricted set on which Assumption 3.1 imposes strong convexity, and the proof of Lemma 3.3 (see Eq. (A15)) uses (8) to replace C1(ΔL1, ΔS1) by 4C1(ΔA1, ΔM_S1). Without a proof of Lemma 3.2, the derivation of the error bound in Lemma 3.3 and Theorem 3.1 is incomplete.
[Appendix A, Proof of Lemma 3.1] The appendix section titled "Proof of Lemma 3.1" does not prove Lemma 3.1. It begins by invoking Assumption 3.1, which is not an assumption of that lemma, and it proceeds to derive the bound stated as Lemma 3.3. The decomposition inequalities (6)–(7) in Lemma 3.1 are never established. As a consequence, the restricted set used in Assumption 3.1 is itself characterized by an unproved lemma, and the proof of the main theorem contains a circular dependency.

minor comments (4)

[Assumption 3.2 (Eq. (10))] Assumption 3.2 is typeset ambiguously: "∥L1∥∞ ≤ α1√d1d1" appears to intend a vanishing spikiness bound such as α1/(√d1 d1), but neither the statement nor the proof (which writes "2α1√d1d1" without a division sign) is clear; this should be corrected for reproducibility.
[Theorem 3.1] Theorem 3.1's displayed bound is garbled in the text: "c1Q2 1 2 logd1 T" should read c1 Q1^2 (2 log d1)/T, and "c2 γ2α2 1 d2 1" should read c2 γ^2 α1^2 / d1^2; a careful proofreading pass is needed.
[Theorem 3.1] The probability statement in Theorem 3.1, 1 − max(e^{−c1 log d1}, e^{−c2 log d2}), does not converge to 1 when d1 and d2 are fixed while T grows; the theorem should state the joint growth conditions under which the bound holds with high probability.
[Section 4.1] In the data-generating process for the sparse case, stationarity is described as being checked by the maximum absolute eigenvalue of the scaled sparse matrix, but for the additive model (1) the stability condition concerns the spectral radius of the corresponding linear operator on the vectorized process; the description should be aligned with the model.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular step in the AMAR error bound; the identifiability gauge shift is a non-circular correctness risk.

full rationale

The central derivation is not circular. Theorem 3.1's bound on e2 is obtained from the basic inequality (A1), the decomposability lemmas, restricted strong convexity (Assumption 3.1), the spikiness/incoherence condition on L1 and L2 (Assumption 3.2), and standard Gaussian concentration on D1 and D2; the target upper bound is the conclusion, not an input. The concentration argument follows Basu and Michailidis [1] and the error decomposition follows Agarwal et al. [22], both external, and no fitted parameter is relabeled as a prediction. The self-citations [28] and [31] are used only for a background remark and for the RMSE convention, so they are not load-bearing. There is, however, a non-circular identifiability gap that affects the interpretation: equation (1) is invariant under (A,B) -> (A+cI_{d1}, B-cI_{d2}), and Assumption 3.2 separates low-rank from sparse components only, not this row/column shift. Thus the recovery statement in Theorem 3.1 is relative to a chosen representative, and the Section 5 reading of L1 and L2 as row-wise and column-wise baseline effects is not uniquely determined by the data. This is a correctness and identification concern, not a derivation-equals-input circularity, so it does not raise the circularity score above 2.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central result rests on several unproved or under-supported premises: two lemmas that characterize the restricted error set (one with an invalid proof, one absent), standard restricted strong convexity and incoherence assumptions, and the implicit identifiability of the additive decomposition. The method itself introduces no new physical entities.

free parameters (3)

lambda_L1, lambda_L2 (nuclear norm penalties) = chosen by grid search / AIC
Regularization weights for low-rank components; selected empirically and influence the bound through lambda_L1^2 R1 + lambda_L2^2 R2.
lambda_S1, lambda_S2 (l1 penalties) = chosen by grid search / AIC
Regularization weights for sparse components; selected empirically and influence the bound through lambda_S1^2 s1 + lambda_S2^2 s2.
alpha1, alpha2 (spikiness parameters) = set between extremes by user
Introduced in Assumption 3.2 to control incoherence between low-rank and sparse components; no data-driven rule given.

assumptions (7)

ad hoc to paper Lemma 3.1: for any rank R1,R2 there exists a decomposition satisfying inequality (6)-(7)
The appendix proof invokes Assumption 3.1, which is ordered after the lemma and does not prove the stated inequality; the result is load-bearing for the restricted set characterization.
ad hoc to paper Lemma 3.2: errors satisfy the error set constraints (8) under lambda lower bounds
Stated without proof in Section 3; needed to define the restricted set on which RSC is assumed.
domain assumption Assumption 3.1: Restricted Strong Convexity with curvature gamma over the restricted set
Standard in high-dimensional M-estimation; assumed rather than verified for the additive MAR loss.
ad hoc to paper Assumption 3.2: ||L1||_inf <= alpha1/(sqrt(d1) d1) and ||L2||_inf <= alpha2/(sqrt(d2) d2)
Incoherence/spikiness condition on the true low-rank components; used to bound cross terms in the proof but applied to Delta L without justification.
domain assumption Assumption 3.3: lambda_L1 >= 4||D1||_sp, lambda_S1 >= 4||D1||_inf + 4 gamma alpha1/(sqrt(d1) d1), and analogous lower bounds for lambda_L2, lambda_S2
Standard lower bounds on regularization parameters in terms of noise-data cross terms.
domain assumption Errors E_t are white noise, and for Theorem 3.1 vec(E_t) are iid Gaussian with covariance Sigma = Sigma1 otimes I + I otimes Sigma2
Defines the stochastic model; the theorem's probability statement depends on Gaussianity.
domain assumption Bounded spectral densities for the Gaussian processes forming E1, Y_-1, E2, Y_-2
Used to control the spectral norms of D1 and D2 in the proof of Theorem 3.1.

how reviews work

0 comments

Cite this review

Pith. "Pith review of High-Dimensional Regularized Additive Matrix Autoregressive Model." pith.science (2026). https://pith.science/paper/CYPFAXVM

@misc{pith2026250601403,
  author       = {Pith},
  title        = {Pith review of: High-Dimensional Regularized Additive Matrix Autoregressive Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYPFAXVM}},
  note         = {Machine review of arXiv:2506.01403}
}

read the original abstract

High-dimensional time series has diverse applications in econometrics and finance. Recent models for capturing temporal dependence have employed a bilinear representation for matrix time series, or the Tucker-decomposition based representation in case of tensor time series. A bilinear or Tucker-decomposition based temporal effect is difficult to interpret on many occasions, along with its computational complexity due to the non-convex nature of the underlying optimization problem. Moreover, the existing matrix case models have not sufficiently explored the possibilities of imposing any lower-dimensional pattern on the transition matrices. In this work, we propose a regularized additive matrix autoregressive model with additive interaction of row-wise and column-wise temporal dependence, that offers more interpretability, less computational burden due to its convex nature and estimation of the underlying low rank plus sparse pattern of its transition matrices. We address the issue of identifiability of the various components in our model and subsequently develop a scalable Alternating Block Minimization algorithm for estimating the parameters. We provide a finite sample error bound under high-dimensional scaling for the model parameters. Finally, the efficacy of the proposed model is demonstrated on synthetic and real data.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

36 extracted references · 33 canonical work pages

[1]

Basu, S., Michailidis, G.: Regularized estimation in sparse high-dimensional time series models (2015)

work page 2015
[2]

Zhang, D., Wu, W.B.: Gaussian approximation for high dimensional time series (2017)

work page 2017
[3]

Journal of the American Statistical Association 117(539), 1338–1356 (2022)

Wang, D., Zheng, Y., Lian, H., Li, G.: High-dimensional vector autoregressive time series modeling via tensor decomposition. Journal of the American Statistical Association 117(539), 1338–1356 (2022)

work page 2022
[4]

Journal of Econometrics 235(2), 1114–1143 (2023)

Adamek, R., Smeekes, S., Wilms, I.: Lasso inference for high-dimensional time series. Journal of Econometrics 235(2), 1114–1143 (2023)

work page 2023
[5]

De Mol, C., Giannone, D., Reichlin, L.: Forecasting using a large number of predictors: Is bayesian shrinkage a valid alternative to principal components? Journal of Econometrics 146(2), 318–328 (2008)

work page 2008
[6]

The Quarterly journal 22 of economics 120(1), 387–422 (2005)

Bernanke, B.S., Boivin, J., Eliasz, P.: Measuring the effects of monetary policy: a factor-augmented vector autoregressive (favar) approach. The Quarterly journal 22 of economics 120(1), 387–422 (2005)

work page 2005
[7]

the Quarterly Journal of economics 117(4), 1329–1368 (2002)

Blanchard, O., Perotti, R.: An empirical characterization of the dynamic effects of changes in government spending and taxes on output. the Quarterly Journal of economics 117(4), 1329–1368 (2002)

work page 2002
[8]

Journal of Multivariate Analysis 170, 232–243 (2019)

Gao, Y., Shang, H.L., Yang, Y.: High-dimensional functional time series fore- casting: An application to age-specific mortality rates. Journal of Multivariate Analysis 170, 232–243 (2019)

work page 2019

Show all 36 references

[9]

Mathematical biosciences 246(2), 326–334 (2013)

Michailidis, G., d’Alch´ e-Buc, F.: Autoregressive models for gene regulatory net- work inference: Sparsity, stability and causality issues. Mathematical biosciences 246(2), 326–334 (2013)

2013
[10]

arXiv preprint arXiv:1901.00769 (2019)

Chen, E.Y., Chen, R.: Modeling dynamic transport network with matrix fac- tor models: with an application to international trade flow. arXiv preprint arXiv:1901.00769 (2019)

2019 arXiv
[11]

Journal of Neuroscience 35(8), 3293–3297 (2015)

Seth, A.K., Barrett, A.B., Barnett, L.: Granger causality analysis in neuroscience and neuroimaging. Journal of Neuroscience 35(8), 3293–3297 (2015)

2015
[12]

Journal of applied Econometrics 25(1), 71–92 (2010)

Ba´ nbura, M., Giannone, D., Reichlin, L.: Large bayesian vector auto regressions. Journal of applied Econometrics 25(1), 71–92 (2010)

2010
[13]

Journal of Econometrics 186(2), 325–344 (2015)

Kock, A.B., Callot, L.: Oracle inequalities for high dimensional vector autoregres- sions. Journal of Econometrics 186(2), 325–344 (2015)

2015
[14]

Journal of the American Statistical Association (2018)

Ghosh, S., Khare, K., Michailidis, G.: High-dimensional posterior consistency in bayesian vector autoregressive models. Journal of the American Statistical Association (2018)

2018
[15]

arXiv preprint arXiv:2101.04276 (2021)

Wang, D., Zheng, Y., Li, G.: High-dimensional low-rank tensor autoregressive time series modeling. arXiv preprint arXiv:2101.04276 (2021)

2021 arXiv
[16]

Journal of Econometrics 222(1), 539–560 (2021)

Chen, R., Xiao, H., Yang, D.: Autoregressive models for matrix-valued time series. Journal of Econometrics 222(1), 539–560 (2021)

2021
[17]

arXiv preprint arXiv:2110.00928 (2021)

Li, Z., Xiao, H.: Multi-linear tensor autoregressive models. arXiv preprint arXiv:2110.00928 (2021)

2021 arXiv
[18]

SIAM review 51(3), 455–500 (2009)

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

2009
[19]

Journal of Business and Economic Statistics (2022)

Xiao, H., Han, Y., Chen, R., Liu, C.: Reduced rank autoregressive models for matrix time series. Journal of Business and Economic Statistics (2022)

2022
[20]

Journal of Computational and Graphical Statistics 30(4), 1143–1155 23 (2021)

Hsu, N.-J., Huang, H.-C., Tsay, R.S.: Matrix autoregressive spatio-temporal models. Journal of Computational and Graphical Statistics 30(4), 1143–1155 23 (2021)

2021
[21]

Journal of Time Series Analysis 45(3), 398–420 (2024)

Zhang, H.-F.: Additive autoregressive models for matrix valued time series. Journal of Time Series Analysis 45(3), 398–420 (2024)

2024
[22]

Agarwal, A., Negahban, S., Wainwright, M.J.: Noisy matrix decomposition via convex relaxation: Optimal rates in high dimensions (2012)

2012
[23]

Machine learning 73(3), 243–272 (2008)

Argyriou, A., Evgeniou, T., Pontil, M.: Convex multi-task feature learning. Machine learning 73(3), 243–272 (2008)

2008
[24]

In: Proceedings of the 24th International Conference on Machine Learning, pp

Tomioka, R., Aihara, K.: Classifying matrices with a spectral regularization. In: Proceedings of the 24th International Conference on Machine Learning, pp. 895– 902 (2007)

2007
[25]

In: Proceedings of the 26th Annual International Conference on Machine Learning, pp

Ji, S., Ye, J.: An accelerated gradient method for trace norm minimization. In: Proceedings of the 26th Annual International Conference on Machine Learning, pp. 457–464 (2009)

2009
[26]

The Journal of Machine Learning Research 17(1), 5097–5147 (2016)

Lin, J., Basu, S., Banerjee, M., Michailidis, G.: Penalized maximum likelihood estimation of multi-layered gaussian graphical models. The Journal of Machine Learning Research 17(1), 5097–5147 (2016)

2016
[27]

Statistical Science 27(4), 538–557 (2012)

Negahban, S.N., Ravikumar, P., Wainwright, M.J., Yu, B., et al.: A unified framework for high-dimensional analysis of m-estimators with decomposable regularizers. Statistical Science 27(4), 538–557 (2012)

2012
[28]

Electronic Journal of Statistics 16(1), 2683–2723 (2022)

Roy, S., Michailidis, G.: Regularized high dimension low tubal-rank tensor regression. Electronic Journal of Statistics 16(1), 2683–2723 (2022)

2022
[29]

IEEE Transactions on Information Theory (2019)

Liu, X.-Y., Aeron, S., Aggarwal, V., Wang, X.: Low-tubal-rank tensor comple- tion using alternating minimization. IEEE Transactions on Information Theory (2019)

2019
[30]

IEEE Transactions on Signal Processing 65(6), 1511–1526 (2016)

Zhang, Z., Aeron, S.: Exact tensor completion using t-svd. IEEE Transactions on Signal Processing 65(6), 1511–1526 (2016)

2016
[31]

Statistica Sinica 33, 1629–1652 (2023)

Chakraborty, N., Khare, K., Michailidis, G.: A bayesian framework for sparse estimation in high-dimensional mixed frequency vector autoregressive models. Statistica Sinica 33, 1629–1652 (2023)

2023
[32]

Technical report, National Bureau of Economic Research (2020)

McCracken, M., Ng, S.: Fred-qd: A quarterly database for macroeconomic research. Technical report, National Bureau of Economic Research (2020)

2020
[33]

Manuscript, Princeton University 46 (2005)

Stock, J.H., Watson, M.W.: An empirical comparison of methods for forecasting using many predictors. Manuscript, Princeton University 46 (2005)

2005
[34]

ˆ∆L1 + ˆ∆S1 2 F + ˆ∆L2 + ˆ∆S2 2 F # . One may note that, γ 2

Wang, D., Liu, X., Chen, R.: Factor models for matrix-valued high-dimensional 24 time series. Journal of econometrics 208(1), 231–248 (2019) Appendix A Proofs of the theoretical results Basic Inequality 1 2T TX t=1 ˆ∆L1 + ˆ∆S1 Yt−1 + Yt−1 ˆ∆L2 + ˆ∆S2 T 2 F ≤ 1 T TX t=1 Et, ˆ∆L...

2019
[35]

So we choose λL2 = k∗ 2Q2 q 2d2 T , for a suitable chosen constant k∗

Following a similar reasoning, it can be shown that P r{ ET 2 Y (2) −1 sp T > 2π k s c12d2 log(1 + 2 ϵ ) cT Q2} ≤6 exp[−c4d2] (A28) for a suitable chosen constant c4. So we choose λL2 = k∗ 2Q2 q 2d2 T , for a suitable chosen constant k∗
[36]

Now the proof of the theorem follows by using these choices of the regularizer parameters and putting the same in the bound obtained in Lemma 3.3. 32

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Basu, S., Michailidis, G.: Regularized estimation in sparse high-dimensional time series models (2015)

work page 2015

[2] [2]

Zhang, D., Wu, W.B.: Gaussian approximation for high dimensional time series (2017)

work page 2017

[3] [3]

Journal of the American Statistical Association 117(539), 1338–1356 (2022)

Wang, D., Zheng, Y., Lian, H., Li, G.: High-dimensional vector autoregressive time series modeling via tensor decomposition. Journal of the American Statistical Association 117(539), 1338–1356 (2022)

work page 2022

[4] [4]

Journal of Econometrics 235(2), 1114–1143 (2023)

Adamek, R., Smeekes, S., Wilms, I.: Lasso inference for high-dimensional time series. Journal of Econometrics 235(2), 1114–1143 (2023)

work page 2023

[5] [5]

De Mol, C., Giannone, D., Reichlin, L.: Forecasting using a large number of predictors: Is bayesian shrinkage a valid alternative to principal components? Journal of Econometrics 146(2), 318–328 (2008)

work page 2008

[6] [6]

The Quarterly journal 22 of economics 120(1), 387–422 (2005)

Bernanke, B.S., Boivin, J., Eliasz, P.: Measuring the effects of monetary policy: a factor-augmented vector autoregressive (favar) approach. The Quarterly journal 22 of economics 120(1), 387–422 (2005)

work page 2005

[7] [7]

the Quarterly Journal of economics 117(4), 1329–1368 (2002)

Blanchard, O., Perotti, R.: An empirical characterization of the dynamic effects of changes in government spending and taxes on output. the Quarterly Journal of economics 117(4), 1329–1368 (2002)

work page 2002

[8] [8]

Journal of Multivariate Analysis 170, 232–243 (2019)

Gao, Y., Shang, H.L., Yang, Y.: High-dimensional functional time series fore- casting: An application to age-specific mortality rates. Journal of Multivariate Analysis 170, 232–243 (2019)

work page 2019