Graph-based Square-Root Estimation for Sparse Linear Regression

Chao Ying; Peili Li; Yunhai Xiao; Zhou Yu; Zhuomei Li

REVIEW 2 major objections 4 minor 54 references

Graph-based Square-Root Estimation for Sparse Linear Regression

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Graph-based square-root estimation makes sparse regression tuning independent of the error standard deviation, with finite-sample bounds, asymptotic normality, and sign consistency.

desk verdict GSRE is a genuinely new estimator with strong empirical results, but the main theorem's central assumption is violated by its own Example 3, and the complete-graph claim in Lemma 2.2(c) is dubious. read the letter →

arxiv 2411.12479 v2 pith:R7NF4LIJ submitted 2024-11-19 stat.ME stat.CO

Peili Li , Zhuomei Li , Yunhai Xiao , Chao Ying , Zhou Yu This is my paper · ORCID

classification stat.MEstat.CO MSC 62J0762F1262J05

keywords sparselinearregressionsquare-rootlossgraphicalstructureamongpredictorsnode-wisegraphpenaltysigma-freetuningalternatingdirectionmethodofmultipliersoraclepropertyhigh-dimensionalstatistics

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 proposes the graph-based square-root estimator (GSRE) for sparse linear regression, which replaces the usual squared loss with the square-root loss $\|y-X\beta\|_2/\sqrt{n}$ and penalizes coefficients with a node-wise graph norm built from each predictor's graph neighborhood. The central aim is to show that this estimator can recover the true sparse signal with finite-sample error bounds, asymptotic normality, and model-selection consistency while choosing the tuning parameter $\lambda$ without knowing or estimating the noise standard deviation $\sigma$. If the claims hold, a practitioner can run sparse regression on correlated predictors under non-Gaussian noise and still get near-oracle estimation and selection. The paper also shows that square-root lasso and group square-root lasso are special cases of the model, and that a complete predictor graph links it to a squared-$\ell_2$-penalized problem.

What carries the argument

The key object is the node-wise graph norm $\|\beta\|_{G,\tau} = \min_{\sum_i V^{(i)}=\beta,\ \mathrm{supp}(V^{(i)})\subseteq N_i}\sum_i \tau_i\|V^{(i)}\|_2$, which lets the graph structure enter as overlapping groups instead of edge pairs. It is coupled with the square-root loss $\|y-X\beta\|_2/\sqrt{n}$, whose homogeneity makes the subgradient condition depend on the ratio $X_{N_i}^\top(y-X\beta)/\|y-X\beta\|_2$ rather than on $\sigma$. The ratio statistic $V=\max_i \sqrt{n}\|X_{N_i}^\top\epsilon\|_2/(\tau_i\|\epsilon\|_2)$ converts the choice of $\lambda$ into a deviation bound on a maximum of Gaussian quadratic forms, removing $\sigma$ from the tuning. The theoretical engine is then a compatibility-type condition on the design adapted to overlapping neighborhoods, plus the neighborhood-containment assumption.

What would settle it

Simulate a design with $p=100$, $s^*=15$, one true predictor connected by an edge to a zero predictor, fit GSRE with the correct graph, and check whether the prediction-error bound $\|X(\hat\beta-\beta^*)\|_2/\sqrt{n}\lesssim \sigma\lambda\sqrt{s^*}/(n\kappa)$ holds across 50 replications; a systematic violation would show Assumption 2.2 is doing the work.

Watch

Extended reading notes

Core claim

On a design with known predictor graph $G$ and weights $\tau_i$, GSRE minimizes $\|y-X\beta\|_2/\sqrt{n} + (\lambda/n)\|\beta\|_{G,\tau}$ with $\|\beta\|_{G,\tau}$ built from overlapping neighborhood groups $N_i$. The paper's main result, Theorem 2.1, states that under the assumption that each true predictor's neighborhood lies inside the true support and a compatibility condition on $X$, every optimal solution obeys $\|X(\hat\beta-\beta^*)\|_2/\sqrt{n} \lesssim \sigma\lambda\sqrt{s^*}/(n\kappa)$, with corresponding bounds for the graph norm and Euclidean error, all on an event that holds with high probability. The tuning parameter is $\sigma$-free: $\lambda$ is set from a maximum over neighborhoods of the ratio $\sqrt{n}\|X_{N_i}^\top\epsilon\|_2/(\tau_i\|\epsilon\|_2)$, so no error-variance estimate enters. For fixed $p$, Theorem 2.2 gives $\sqrt{n}(\hat\beta_{I^*}-\beta^*_{I^*}) \xrightarrow{d} N(0,\sigma^2 M_{I^*,I^*}^{-1})$, and in high dimensions Theorem 2.3 gives sign consistency under an irrepresentable-type condition.

Load-bearing premise

The load-bearing premise is Assumption 2.2: every predictor that is truly useful must have all of its graph neighbors also truly useful; if a signal variable is connected to a noise variable, the finite-sample bounds and selection guarantees in the paper do not apply.

Editorial extensions

If this is right

A user can calibrate $\lambda$ from $n$, the graph degrees, and a confidence level $\alpha$ alone; no preliminary estimate of the noise variance is needed.
The error bounds and selection consistency extend to non-Gaussian noise settings such as $t$, Laplace, and uniform errors, matching the experiments in the paper.
For fixed dimension, coefficient estimates on the true support are asymptotically normal with the oracle covariance matrix $\sigma^2 M_{I^*,I^*}^{-1}$, enabling standard inferential readouts.
Under a graph-adapted irrepresentable condition, the estimator recovers the exact sign pattern with probability tending to one even when $p$ grows with $n$.
Because square-root lasso and group square-root lasso are special cases of GSRE, any algorithmic or theoretical improvement for GSRE transfers to those established estimators.

Reading between the lines

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

Beyond the paper: the neighborhood-containment condition is the real design constraint; if a useful predictor is adjacent to a pure noise predictor, the node-wise penalty will tend to bring that noise variable into the support, so the method's practical safety depends on graph estimation quality.
Beyond the paper: the paper only tests sensitivity to graph misspecification in the supplement; an explicit extension would derive bounds under partial neighborhood contamination, allowing a fraction of $N_i \cap I^*_c$ nodes.
Beyond the paper: the $\sigma$-free tuning mechanism suggests a natural plug-in rule for adaptive variants of GSRE with nonconvex penalties, where the same ratio statistic could set local penalty scales without cross-validation.
Beyond the paper: on complete graphs the nonzero solutions coincide with a square-root-loss plus squared-$\ell_2$ penalty, so the graph penalty is doing ridge-like shrinkage; a user should expect edge density to change selection behavior.

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.

Desk Editor's Note

read the letter

Quick take: GSRE is a genuinely new estimator — the first square-root-loss variant of Yu and Liu's node-wise graph penalty — with sigma-free tuning, sensible theory, and strong simulations across noise types. Worth taking seriously. But the main finite-sample theorem is proven under an assumption that the paper's own flagship example violates, and Lemma 2.2(c) looks wrong.

What's new and good: the combination is natural and the paper is the first to present it. The special-case reductions to square-root lasso and group square-root lasso are correctly derived. The sigma-free tuning bound (Lemma 2.3) is a real advantage over the original SRIG, whose tuning depends on σ. The ADMM algorithm with the Woodbury identity is standard but sound. The simulations are extensive: three predictor structures, four noise types, p=100, n=20/40/60, and GSRE generally wins on L2 error, prediction, and MCC. The two real-data examples also favor it.

Soft spots, in order of seriousness. First, Assumption 2.2 — every active predictor's neighborhood must lie inside the true support — is load-bearing for Theorem 2.1 and Theorem 2.3. Example 3 violates it: with AR(1) design and β* supported on {1,...,15}, node 15 is adjacent to inactive node 16. So the main bound doesn't cover the paper's most impressive simulation. The sensitivity study is relegated to the supplement, and Remark 2.5's "reasonable" doesn't help. This is a coverage gap, not a fatal flaw, but it should be fixed or clearly acknowledged. Second, Lemma 2.2(c): for a complete graph, the node-wise norm is a constant times ||β||_2, not ||β||_2^2. Claiming the GSRE problem has the same nonzero solutions as a squared-L2-penalized problem is at best trivial (both are dense) and at worst false. This needs correction. Third, the non-Gaussian claim: the finite-sample bounds rely on Gaussian noise via Lemma 2.3; Remark 2.4 only sketches sub-exponential extensions. The simulations do show robustness, but the theoretical statement should be toned down or backed with a real proof.

Verdict: this deserves peer review, not a desk reject. I'd send it to a capable referee with instructions to push on Assumption 2.2 and Lemma 2.2(c). The estimator itself is useful and the empirical story is convincing; the theory needs tightening.

Referee Report

2 major / 4 minor

Summary. The paper proposes graph-based square-root estimation (GSRE), a sparse linear regression estimator that combines the square-root loss with a node-wise graphical penalty. The authors show that the method reduces to square-root lasso and group square-root lasso for special graphs, provide finite-sample oracle-type bounds, asymptotic normality, and model-selection consistency under a σ-free tuning parameter, and present an ADMM algorithm. Numerical experiments on simulated and real data compare GSRE with lasso, adaptive lasso, elastic net, square-root lasso, and graph-based least-squares regression. The main theoretical contribution is that the tuning parameter does not depend on the unknown noise level σ, and the main practical claim is improved estimation, prediction, and variable selection under heavy-tailed and non-Gaussian noise.

Significance. If the theoretical claims are correct, the paper makes a useful extension of square-root lasso to graph-structured penalties, and the σ-free tuning construction is a genuine practical advantage in high-dimensional settings. The paper also ships code and reports extensive simulations across several noise distributions and predictor graphs, which strengthens the empirical contribution. However, the two issues raised below — the violation of the key graph-support assumption by the paper's own flagship example, and an incorrect equivalence claim for complete graphs — need to be addressed before the theoretical and generality claims can be taken at face value.

major comments (2)

[Section 2.4 (Assumption 2.2) and Section 4.1 (Example 3)] Example 3 of Section 4.1 violates Assumption 2.2, which is required by Theorem 2.1 and Theorem 2.3. In Example 3 the predictors follow an AR(1) model with Σ_{ij} = 0.5^{|i-j|}, so the graph is a path with edges between consecutive nodes. With β* supported on {1,...,15}, node 15 has neighborhood {14,15,16}, and node 16 is inactive. Hence Assumption 2.2 fails for i=15, and the finite-sample bounds and model-selection consistency results do not cover this simulation, which is one of the paper's main advertised successes. The main text only states that a sensitivity analysis appears in the supplementary materials, which does not repair the coverage gap. Please either extend the theory to allow active nodes adjacent to inactive nodes, or explicitly state that the theoretical guarantees apply only when Assumption 2.2 holds and classify Example 3 as outside the theoretical scope.
[Lemma 2.2(c)] The claimed equivalence in Lemma 2.2(c) is not correct. For a complete graph, every neighborhood N_i equals [p], so the graph norm ∥β∥_{G,τ} equals (min_i τ_i) ∥β∥_2: putting all mass in the node with the smallest τ_i gives the upper bound, and the triangle inequality gives the lower bound. Since ∥β∥_{G,τ} is positively homogeneous while the squared ℓ2 penalty in (7) is not, the two optimization problems cannot share the same solutions. This invalidates the assertion that GSRE reduces to the squared-ℓ2-penalized problem (7) when the predictor graph is complete. Please correct or remove this part, and revisit the 'generality' discussion in Remark 2.2 accordingly.

minor comments (4)

[Remark 2.6] The claim that 'subset recovery can be guaranteed with no noise, i.e., σ = 0' is problematic because the event A in (8) is defined through the ratio V = max_i √n ∥X_{N_i}^⊤ ε∥₂ / (τ_i ∥ε∥₂), which is undefined when ε = 0. The σ = 0 case should be treated separately or the statement qualified.
[Section 3] The augmented-Lagrangian step size is denoted by σ, which clashes with the noise-level σ used throughout the rest of the paper. Please use a different symbol, e.g., ρ, to avoid confusion.
[Section 4.1] There is a typo in the definition of the Matthews correlation coefficient: 'Mattews' should be 'Matthews'. Also, the caption of Figure 3 refers to 'bodyfa2' while the text uses 'bodyfat2'.
[Notation and Section 2.4] The notation '≲' used in Theorem 2.1 is not defined in the paper. Please specify whether it hides absolute constants, and clarify the dependence of the constants on the model parameters.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the GSRE bounds, asymptotics, and consistency results are derived from stated assumptions with a sigma-free tuning parameter, and the only self-citation is a non-load-bearing literature pointer.

full rationale

The paper's derivation chain is self-contained. The tuning parameter lambda in Lemma 2.3 is expressed directly in terms of n, p, alpha, graph degrees, and operator norms (Equation 9), with no dependence on sigma or on the fitted coefficients, so the finite-sample bounds in Theorem 2.1 are not fitted inputs relabeled as predictions. The bounds are standard oracle-type and group-lasso-style inequalities, and the asymptotic normality and model-selection consistency results (Theorems 2.2 and 2.3) are proved under explicit assumptions that do not presuppose the conclusions. Lemma 2.2's equivalences are algebraic identities comparing the GSRE objective to square-root lasso, group square-root lasso, and a squared-L2 penalized problem; they follow from the graph structures rather than renaming the paper's own results. The numerical section is an empirical benchmark against external methods (lasso, ridge, elastic net, square-root lasso, and SRIG), and no experiment is presented as a derivation. The only self-citation, Ding et al. (2023) cited among 'more research on the square-root-loss function,' is a literature pointer and carries no argumentative weight. Assumption 2.2 restricts neighborhoods of active predictors to the true support; whether the paper's Example 3 satisfies this condition is a correctness or coverage question, not a circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard high-dimensional assumptions plus Assumption 2.2, which is restrictive. The theory is Gaussian-only; non-Gaussian results are deferred to future work or the supplementary material.

free parameters (1)

tuning parameter lambda = grid search over 2^-13 to 2^-5 times sqrt(n/2)/||X||, selected by HBIC
lambda controls the trade-off between square-root loss and graph penalty. Theoretically it is set by inequality (9) depending on alpha, but in experiments it is selected by a data-dependent criterion.

assumptions (6)

domain assumption Assumption 2.1: The additive noise terms follow a standard Gaussian distribution.
Used in Lemma 2.3 to construct the sigma-free lambda and in Theorem 2.1 for the finite-sample bounds.
domain assumption Assumption 2.2: For each i in I*, the neighborhood N_i is a subset of I*.
This strong condition ensures the graph penalty does not pull noise predictors into the support. It is load-bearing for the oracle bounds and may fail in real graphs.
domain assumption Assumption 2.3: A compatibility condition with constant kappa holds for the design matrix.
This is a standard restricted-eigenvalue-type condition used to obtain the oracle bounds in Theorem 2.1.
domain assumption Assumption 2.4: The sparsity s* satisfies s* < n^2 kappa^2 / lambda^2.
This keeps the true model size small relative to the sample size and the tuning parameter.
domain assumption Assumptions 2.7-2.10: sparsity, bounded predictors, lower eigenvalue on the true support, and an irrepresentable-type condition.
These are standard high-dimensional assumptions for model selection consistency in Theorem 2.3.
domain assumption When the graph G is unknown, the graphical lasso estimate of the precision matrix is accurate enough.
The theory assumes G is known; in experiments G is estimated. The simulations show GSRE matches GSRE-o, but this is not covered by the theorems.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Graph-based Square-Root Estimation for Sparse Linear Regression." pith.science (2026). https://pith.science/paper/R7NF4LIJ

@misc{pith2026241112479,
  author       = {Pith},
  title        = {Pith review of: Graph-based Square-Root Estimation for Sparse Linear Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7NF4LIJ}},
  note         = {Machine review of arXiv:2411.12479}
}

read the original abstract

Sparse linear regression is one of the classic problems in the field of statistics, which has deep connections and high intersections with optimization, computation, and machine learning. To address the effective handling of high-dimensional data, the diversity of real noise, and the challenges in estimating standard deviation of the noise, we propose a novel and general graph-based square-root estimation (GSRE) model for sparse linear regression. Specifically, we use square-root-loss function to encourage the estimators to be independent of the unknown standard deviation of the error terms and design a sparse regularization term by using the graphical structure among predictors in a node-by-node form. Based on the predictor graphs with special structure, we highlight the generality by analyzing that the model in this paper is equivalent to several classic regression models. Theoretically, we also analyze the finite sample bounds, asymptotic normality and model selection consistency of GSRE method without relying on the standard deviation of error terms. In terms of computation, we employ the fast and efficient alternating direction method of multipliers. Finally, based on a large number of simulated and real data with various types of noise, we demonstrate the performance advantages of the proposed method in estimation, prediction and model selection.

Figures

Figures reproduced from arXiv: 2411.12479 by the authors.

**Figure 2.** Estimated graphs of real data. to estimate the graphical structure of the predictors, as detailed in [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** Comparison of MSE for various methods on real data. 5 Conclusion In this paper, we addresses the high-dimensional sparse linear regression problem by proposing a graph-based square-root estimation (GSRE) method, which integrates square-root loss and node-wise graph penalty. This paper highlights the broad applicability of the proposed method by analyzing its equivalence with several classical square-root estimation… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 48 canonical work pages

[1]

Beck, A. (2017). First-order methods in optimization . SIAM

2017
[2]

Bellec, P. C., G. Lecu \'e , and A. B. Tsybakov (2018). Slope meets lasso: improved oracle bounds and optimality. The Annals of Statistics\/ 46\/ (6B), 3603--3642

work page 2018
[3]

Chernozhukov, and L

Belloni, A., V. Chernozhukov, and L. Wang (2011). Square-root lasso: pivotal recovery of sparse signals via conic programming. Biometrika\/ 98\/ (4), 791--806

work page 2011
[4]

Bickel, P. J., Y. Ritov, and A. B. Tsybakov (2009). Simultaneous analysis of L asso and D antzig selector. The Annals of Statistics\/ 37\/ (4), 1705--1732

work page 2009
[5]

Bondell, H. D. and B. J. Reich (2008). Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR . Biometrics\/ 64\/ (1), 115--123

work page 2008
[6]

Bousquet, O. (2002). A B ennett concentration inequality and its application to suprema of empirical processes. Comptes Rendus Mathematique\/ 334\/ (6), 495--500

work page 2002
[7]

B \"u hlmann, P. and S. V. D. Geer (2011). Statistics for high-dimensional data: methods, theory and applications . Springer Science & Business Media

work page 2011
[8]

Lederer, and Y

Bunea, F., J. Lederer, and Y. She (2013). The group square-root lasso: T heoretical properties and fast algorithms. IEEE Transactions on Information Theory\/ 60\/ (2), 1313--1325

work page 2013

Show all 54 references

[9]

Liu, and X

Cai, T., W. Liu, and X. Luo (2011). A constrained _1 minimization approach to sparse precision matrix estimation. Journal of the American Statistical Association\/ 106\/ (494), 594--607

2011
[10]

Combettes, P. L. and J.-C. Pesquet (2011). Proximal splitting methods in signal processing. Fixed-point algorithms for inverse problems in science and engineering\/ , 185--212

2011
[11]

Pang, and B

Cui, Y., J.-S. Pang, and B. Sen (2018). Composite difference-max programs for modern statistical estimation problems. SIAM Journal on Optimization\/ 28\/ (4), 3344--3374

2018
[12]

Derumigny, A. (2018). Improved bounds for square-root lasso and square-root slope. Electronic Journal of Statistics\/ 12\/ (1), 741--766

2018
[13]

Zhang, P

Ding, Y., H. Zhang, P. Li, and Y. Xiao (2023). An efficient semismooth N ewton method for adaptive sparse signal recovery problems. Optimization Methods and Software\/ 38\/ (2), 262--288

2023
[14]

Fan, J. and R. Li (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association\/ 96\/ (456), 1348--1360

2001
[15]

Fazel, M., T. K. Pong, D. Sun, and P. Tseng (2013). Hankel matrix rank minimization with applications to system identification and realization. SIAM Journal on Matrix Analysis and Applications\/ 34\/ (3), 946--977

2013
[16]

Frank, L. E. and J. H. Friedman (1993). A statistical view of some chemometrics regression tools. Technometrics\/ 35\/ (2), 109--135

1993
[17]

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
[18]

Friedman, J. H. (1991). Multivariate adaptive regression splines. The Annals of Statistics\/ 19\/ (1), 1--67

1991
[19]

Garzon, R., G. A. Calin, and C. M. Croce (2009). Micro RNA s in cancer. Annual Review of Medicine\/ 60 , 167--179

2009
[20]

Golub, G. H. and C. F. V. Loan (2013). Matrix computations . JHU press

2013
[21]

Hayes, J., P. P. Peruzzi, and S. Lawler (2014). Micro RNA s in cancer: biomarkers, functions and therapy. Trends in Molecular Medicine\/ 20\/ (8), 460--469

2014
[22]

Huang, L., J. Jia, B. Yu, B.-G. Chun, P. Maniatis, and M. Naik (2010). Predicting execution time of computer programs using sparse polynomial regression. Advances in Neural Information Processing Systems\/

2010
[23]

Kim, S. and E. P. Xing (2009). Statistical estimation of correlated genome associations to a quantitative trait network. PLoS Genetics\/ 5\/ (8), e1000587

2009
[24]

Koltchinskii, V. and M. Yuan (2010). Sparsity in multiple kernel learning. The Annals of Statistics\/ 38\/ (6), 3660--3695

2010
[25]

Kong, Y. W., D. Ferland-McCollough, T. J. Jackson, and M. Bushell (2012). Micro RNA s in cancer management. The Lancet Oncology\/ 13\/ (6), e249--e258

2012
[26]

Lederer, J. and S. van de Geer (2011). New concentration inequalities for empirical processes. Bernoulli\/

2011
[27]

Li, C. and H. Li (2008). Network-constrained regularization and variable selection for analysis of genomic data. Bioinformatics\/ 24\/ (9), 1175--1182

2008
[28]

Yu, and Y

Liu, J., G. Yu, and Y. Liu (2019). Graph-based sparse linear discriminant analysis for high-dimensional classification. Journal of Multivariate Analysis\/ 171 , 250--269

2019
[29]

Pontil, S

Lounici, K., M. Pontil, S. V. D. Geer, and A. B. Tsybakov (2011). Oracle inequalities and optimal inference under group sparsity. The Annals of Statistics\/ 39\/ (4), 2164--2204

2011
[30]

Luo, Y. and Y. Liu (2022). Recovery of sums of sparse and dense signals by incorporating graphical structure among predictors. Canadian Journal of Statistics\/ 50\/ (2), 471--490

2022
[31]

Meier, L., S. V. de Geer, and P. B \"u hlmann (2009). High-dimensional additive modeling. The Annals of Statistics\/ 37\/ (6B), 3779--3821

2009
[32]

Jacob, and J.-P

Obozinski, G., L. Jacob, and J.-P. Vert (2011). Group lasso with overlaps: the latent group lasso approach. arXiv preprint arXiv:1110.0413\/

2011 arXiv
[33]

Owen, A. B. (2007). A robust hybrid of lasso and ridge regression. Contemporary Mathematics\/ 443\/ (7), 59--72

2007
[34]

Pan, Y., X. Zhao, S. Wei, and Z. Liu (2023). High-dimensional expectile regression incorporating graphical structure among predictors. Journal of Statistical Computation and Simulation\/ 93\/ (2), 231--248

2023
[35]

Shen, X., L. Chen, Y. Gu, and H.-C. So (2016). Square-root lasso with nonconvex regularization: A n ADMM approach. IEEE Signal Processing Letters\/ 23\/ (7), 934--938

2016
[36]

Stephenson, M., R. A. Ali, G. A. Darlington, F. S. Schenkel, and E. J. Squires (2021). DSLRIG : L everaging predictor structure in logistic regression. Communications in Statistics-Simulation and Computation\/ 50\/ (6), 1600--1612

2021
[37]

Tamayo, V

Subramanian, A., P. Tamayo, V. K. Mootha, et al. (2005). Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles. Proceedings of the National Academy of Sciences\/ 102\/ (43), 15545--15550

2005
[38]

Tang, P., C. Wang, D. Sun, and K.-C. Toh (2020). A sparse semismooth N ewton based proximal majorization-minimization algorithm for nonconvex square-root-loss regression problems. Journal of Machine Learning Research\/ 21\/ (226), 1--38

2020
[39]

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

1996
[40]

van de Geer, S. and J. Lederer (2013). The B ernstein- O rlicz norm and deviation inequalities. Probability Theory and Related Fields\/ 157\/ (1), 225--250

2013
[41]

Rosasco, S

Villa, S., L. Rosasco, S. Mosci, and A. Verri (2014). Proximal methods for the latent group lasso penalty. Computational Optimization and Applications\/ 58 , 381--407

2014
[42]

Wang, L. (2013). The L 1 penalized LAD estimator for high dimensional linear regression. Journal of Multivariate Analysis\/ 120 , 135--151

2013
[43]

Kim, and R

Wang, L., Y. Kim, and R. Li (2013). Calibrating non-convex penalized regression in ultra-high dimension. The Annals of Statistics\/ 41\/ (5), 2505--2536

2013
[44]

Wang, Z., X. Liu, W. Tang, and Y. Lin (2021). Incorporating graphical structure of predictors in sparse quantile regression. Journal of Business & Economic Statistics\/ 39\/ (3), 783--792

2021
[45]

Ching, and M

Wen, Y.-W., W.-K. Ching, and M. Ng (2018). A semi-smooth N ewton method for inverse problem with uniform noise. Journal of Scientific Computing\/ 75\/ (2), 713--732

2018
[46]

Xiu, X., L. Kong, Y. Li, and H. Qi (2018). Iterative reweighted methods for _1- _p minimization. Computational Optimization and Applications\/ 70\/ (1), 201--219

2018
[47]

Yuan, Y.-C

Yang, S., L. Yuan, Y.-C. Lai, X. Shen, P. Wonka, and J. Ye (2012). Feature grouping and selection over an undirected graph. In Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data mining , pp.\ 922--930

2012
[48]

Yu, G. and Y. Liu (2016). Sparse regression incorporating graphical structure among predictors. Journal of the American Statistical Association\/ 111\/ (514), 707--720

2016
[49]

Yuan, M. and Y. Lin (2006). Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society: Series B (Methodological)\/ 68\/ (1), 49--67

2006
[50]

Yuan, M. and Y. Lin (2007). Model selection and estimation in the G aussian graphical model. Biometrika\/ 94\/ (1), 19--35

2007
[51]

Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics\/ 38\/ (2), 894--942

2010
[52]

Zhang, Z. and W. Wei (2015). Primal-dual approach for uniform noise removal. In First International Conference on Information Science and Electronic Technology (ISET 2015) , pp.\ 103--106

2015
[53]

Zhao, P. and B. Yu (2006). On model selection consistency of L asso. The Journal of Machine Learning Research\/ 7 , 2541--2563

2006
[54]

Zhou, and B

Zhou, S., J. Zhou, and B. Zhang (2019). High-dimensional generalized linear models incorporating graphical structure among predictors. Electronic Journal of Statistics\/ 13\/ (2), 3161--3194

2019

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Beck, A. (2017). First-order methods in optimization . SIAM

2017

[2] [2]

Bellec, P. C., G. Lecu \'e , and A. B. Tsybakov (2018). Slope meets lasso: improved oracle bounds and optimality. The Annals of Statistics\/ 46\/ (6B), 3603--3642

work page 2018

[3] [3]

Chernozhukov, and L

Belloni, A., V. Chernozhukov, and L. Wang (2011). Square-root lasso: pivotal recovery of sparse signals via conic programming. Biometrika\/ 98\/ (4), 791--806

work page 2011

[4] [4]

Bickel, P. J., Y. Ritov, and A. B. Tsybakov (2009). Simultaneous analysis of L asso and D antzig selector. The Annals of Statistics\/ 37\/ (4), 1705--1732

work page 2009

[5] [5]

Bondell, H. D. and B. J. Reich (2008). Simultaneous regression shrinkage, variable selection, and supervised clustering of predictors with OSCAR . Biometrics\/ 64\/ (1), 115--123

work page 2008

[6] [6]

Bousquet, O. (2002). A B ennett concentration inequality and its application to suprema of empirical processes. Comptes Rendus Mathematique\/ 334\/ (6), 495--500

work page 2002

[7] [7]

B \"u hlmann, P. and S. V. D. Geer (2011). Statistics for high-dimensional data: methods, theory and applications . Springer Science & Business Media

work page 2011

[8] [8]

Lederer, and Y

Bunea, F., J. Lederer, and Y. She (2013). The group square-root lasso: T heoretical properties and fast algorithms. IEEE Transactions on Information Theory\/ 60\/ (2), 1313--1325

work page 2013