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REVIEW 3 major objections 6 minor 59 references

Guidelines for LASSO and derivatives use under different dependence and scale structures

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper argues that the best LASSO-type procedure depends jointly on dependence structure, scale structure, and the practitioner's objective, and that univariate standardization is not uniformly beneficial.

desk verdict A careful, useful simulation study on when univariate standardization helps or hurts LASSO under dependence and scale; the main caveat is that the guidelines generalize beyond Toeplitz designs without evidence. read the letter →

arxiv 2506.08582 v2 pith:V4ZBKVSG submitted 2025-06-10 stat.ME

Laura Freijeiro-González , Manuel Febrero-Bande , Wenceslao González-Manteiga This is my paper
classification stat.ME MSC 62J07
keywords LASSOvariableselectionunivariatestandardizationdependencestructurescaleeffectshigh-dimensionalregressionpredictionsimulationstudy
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 asks a practical question the LASSO literature leaves mostly open: when covariates are both dependent and on different scales, should you standardize them before penalized regression, and which method should you use? Using Gaussian simulations with diagonal, Toeplitz, and scale-mixed Toeplitz designs, it shows that univariate standardization can break dependence relations and add noise, while raw data can be misled by high-scale noisy covariates. The authors' central claim is that the best procedure is scenario-dependent: methods that maximize true positive recovery differ from methods that minimize false discoveries, and prediction-oriented choices differ from selection-oriented ones. The paper consolidates this into a decision tree with recommendations for p>n and n>p settings, and illustrates the patterns on three real datasets.

What carries the argument

The machinery is the side-by-side comparison of the raw centered design matrix Xr and its univariate standardized version Xus across simulated Gaussian designs whose covariance is diagonal or Toeplitz, with relevant covariates placed in blocks or spread sparsely and with scale variations applied to relevant or noisy covariates. In the orthogonal case, the paper shows that selection depends on the covariate variance and the inner product with the standardized covariate, making scale an explicit selection driver; in non-orthogonal designs no closed-form LASSO solution exists, so dependence and scale interact through the correlation structure. The final decision tree translates the simulation outcomes into method recommendations indexed by dependence (yes/no), scale differences (yes/no), and objective (false discovery proportion, true positive proportion, good-interval MSE, or overestimation correction).

What would settle it

Simulate or collect data with a dependence structure outside the Toeplitz family — for example, a block-diagonal or factor covariance with non-Gaussian errors and the same scale patterns as RNC.TOEP-S — and compare the decision tree's recommendation against raw and standardized LASSO variants; if the recommended method is not the best in selection or prediction, the guidelines fail.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that univariate standardization is not a harmless preprocessing step: it changes which covariates the LASSO selects, and the direction of the change is governed by the dependence-scale configuration. In the orthogonal IND scenarios, raw and standardized data behave similarly, and standardization chiefly reduces scale-driven false discoveries among noisy high-variance covariates. In the unit-scale Toeplitz scenarios (UTOEP-B, UTOEP-S) the two preprocessing choices again behave similarly, with strong correlation (rho=0.9) making a conservative subset-selection strategy competitive with full recovery. In the scale-mixed Toeplitz scenarios (RC.TOEP-S, RNC.TOEP-S) the two failure modes separate: raw-data LASSO is confused by high-scale noisy covariates, while univariate standardization disrupts the dependence structure and adds more noise at a similar rate of relevant-covariate recovery. The authors conclude that method choice should be driven jointly by dependence structure, scale structure, sample regime, and the objective of selection versus prediction.

Load-bearing premise

The load-bearing premise is that the Gaussian diagonal and Toeplitz simulation scenarios are representative enough that the decision tree transfers to arbitrary high-dimensional linear regression problems with dependence and scale variation.

Editorial extensions

If this is right

  • Under strong Toeplitz dependence, methods that exploit the dependence structure (AdapL.1se, Dant, DC.VS) can select a small representative subset of relevant covariates with few false discoveries, instead of forcing full recovery of the true support.
  • In scale-mixed Toeplitz settings, raw-data LASSO tends to be confused by noisy high-scale covariates, and univariate standardization does not remove the confusion: it disrupts dependence and adds more noise while recovering about the same number of relevant terms.
  • Prediction-oriented users should not expect LASSO-type methods to report honest accuracy: most overestimate predictive performance, and only some variants (notably LASSO.1se and LASSO.BIC at n=300) fall into the good-MSE interval in the studied scenarios.
  • Under independence, univariate standardization is mainly a safeguard against scale-driven false discoveries, and raw and standardized selections are otherwise close.
  • The paper's decision tree offers a concrete starting point: choose the method according to dependence (yes/no), scale differences (yes/no), sample regime (p>n or n>p), and whether the goal is minimizing false discoveries, maximizing true positives, obtaining good prediction error, or correcting overestimation.

Reading between the lines

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

  • A natural extension the paper does not pursue is a quantitative threshold for when standardization hurts: the simulations suggest the harm grows with the correlation between relevant and noisy covariates and with the scale gap, but no formula is given.
  • Because the scenarios are confined to Gaussian diagonal/Toeplitz designs, the decision tree's transfer to block-structured or factor-model dependence, non-Gaussian errors, or data with unknown support locations remains untested; applying the tree as a general recipe would be an extrapolation.
  • Practitioners could use the paper's results to justify a stability check: fit both raw and standardized versions on a subsample and compare selection agreement, since either preprocessing can win depending on the configuration.
  • The evidence that a few high-correlation covariates can explain most variability suggests that the notion of 'true support' may be less decision-relevant under strong dependence than a parsimonious predictive set, which is a perspective the paper raises implicitly in its representative-subset discussion.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper investigates the effect of univariate standardization on the LASSO and several competitors (adaptive LASSO, SCAD, Dantzig selector, relaxed LASSO, square-root LASSO, scaled LASSO, and distance-covariance variable selection) in high-dimensional linear regression settings where covariates exhibit dependence and heterogeneous scales. A simulation study with p=100, n=25-300, and covariance structures that are diagonal, Toeplitz (rho=0.5, 0.9), and Toeplitz with added scale perturbations compares raw versus univariately standardized data on two criteria: variable selection (false discovery proportion and true positive proportion) and prediction (MSE relative to a 'Good Interval' and overestimation behavior). The paper concludes with a decision tree (Figure 14) recommending procedures by dependence/scale configuration and objective, and applies the methods to three real datasets.

Significance. The central qualitative finding is useful and convincingly demonstrated within the simulated designs: univariate standardization is not uniformly beneficial, because in dependence settings it can disrupt the dependence structure and add noise, while raw data are vulnerable to large-scale noisy covariates. The paper's strengths include the breadth of competitors, the explicit separation of selection and prediction objectives, the availability of code on GitHub, and the honest acknowledgment in Figure 14's footnote that the recommendation depends on the type of dependence structure. If the scope were appropriately restricted and the real-data prediction protocol corrected, the guidelines could serve as a practical reference. However, as written, the external validity of the decision tree beyond the simulated covariance families is not established, and the real-data predictive comparisons are compromised by the evaluation protocol.

major comments (3)
  1. [Section 7 / Figure 14] The decision tree dichotomizes 'dependence' (Yes/No) and 'scales' (Yes/No) and then gives concrete procedure recommendations for every branch, but the simulation evidence covers only diagonal and Toeplitz Gaussian designs (Section 3) with fixed support locations and coefficient magnitudes. The footnote '(*) It depends on the type of dependence structure' concedes this limitation, but the tree still presents the recommendations as applying to all dependence structures. Since the LASSO's selection behavior under dependence is governed by quantities like the irrepresentable condition and coherence, which can vary substantially within the 'dependence = Yes' category, the guidelines need either to be explicitly restricted to Toeplitz-type dependence with a clear operational definition, or to be validated on additional covariance families (e.g., equicorrelation, factor models, or negative-correlation structures). As written, the title and Section 7 overstate the scope of the findings.
  2. [Sections 6.1-6.3] The real-data prediction protocol selects covariates using the full dataset before splitting into training and test sets; for example, Section 6.1 states: 'First, the relevant covariates are selected employing the total n = 71 samples. Subsequently, the sample is randomly divided in two...'. This is a data-leakage protocol: the test set influences covariate selection, so the reported MSE and %Dev values in Tables 11-13 are optimistic and the comparisons among methods are not honest out-of-sample evaluations. The selection step should be nested inside the training folds (e.g., repeated cross-validation) or performed on the training portion only for each split. Because the real-data analysis is a stated contribution of the paper, this protocol issue is load-bearing and should be corrected.
  3. [Sections 4-5 (all simulation tables)] All simulation results are reported as point estimates over M=500 replicates without standard errors or confidence intervals. For example, in Table 5 the RAW versus UNIV MSE differences for LASSO.1se (12.624 vs 7.092 in RC.IND; 12.366 vs 6.806 in RNC.IND) are large, but in Table 1 differences such as 1.345 vs 1.346 are clearly negligible; without a measure of Monte Carlo error the reader cannot assess which differences are meaningful. Given that the guidelines in Figure 14 aggregate across many scenario cells, the authors should report Monte Carlo standard errors or display variability (e.g., boxplots) for at least the key scenarios. This would substantially strengthen the evidential basis for the qualitative recommendations.
minor comments (6)
  1. [Section 1] The text says 'an independent and identically distributed (idd) sample'; 'idd' should be 'iid'.
  2. [Section 4] The typo 'Topelitz covariance structure' should read 'Toeplitz covariance structure'.
  3. [Section 6.3] Table 13's caption says 'training sample of size 191' but the text says the training sample is 1215 observations; the caption should match the narrative.
  4. [Section 6.1] The sentence 'Results for Xus are displayed in Table 11, while those concerning Xr are collected in Table 11 in Section E.4.1' has a duplicate table number; the Xr table is presumably Table 27 in the appendix and should be cited correctly.
  5. [Section 2, Eq. (2)] The statement 'keeping just the orthogonal assumption of X' is ambiguous because Eq. (2) also assumes the columns of X have equal norm; the authors should explicitly state that the formula holds for orthogonal (not necessarily unit-norm) columns and define sigma_j^2 as the column norm squared.
  6. [Section 3] The condition written as 'n > log(p)s ≈ 4.61s' is imprecise; it should be expressed as n > s log p (with the relevant constant) and the associated references for the consistent-selection condition should be stated more carefully.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: guidelines are empirical summaries tested on real data; self-citations are contextual only.

full rationale

This paper is an empirical simulation study rather than a derivation, and the only closed-form result, equation (2), is a standard orthonormal-LASSO formula cited to Giraud (2014); it is used only to motivate the scale effect and is not the source of the later guidelines. The recommendations in Figure 14 are explicit post hoc summaries of the Monte Carlo results over the Section 3 grid, with the footnote '(*) It depends on the type of dependence structure' acknowledging the limits of the dependence branch, so the tree is not a prediction that reduces to its own inputs by construction. No parameter is fitted to one part of the data and then renamed a prediction; the oracle MSE and GI comparisons are independent evaluation criteria against known simulation truth, and the three real datasets supply an external, albeit qualitative, check. The repeated citations to Freijeiro-González et al. (2022) and Febrero-Bande et al. (2019) are for background, implementation details, and competitor definitions; the Xr-versus-Xus comparisons are computed in this paper using glmnet and are not justified by those citations. The main risk is external validity of the Gaussian diagonal/Toeplitz scenario grid, which is a generality limitation rather than a circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

No new entities are postulated. The paper's contribution relies on hand-chosen simulation parameters and standard modeling assumptions rather than invented mechanisms; the free-parameter list reflects those design choices that drive the conclusions.

free parameters (5)
  • Signal coefficients beta = 1.25 (IND), 0.5 (UTOEP and TOEP-S)
    Chosen by hand in Section 3 to satisfy the beta-min signal recovery condition; recovery rates depend on these magnitudes.
  • Scale values in design covariance = 0.5, 1, 3, 10, 25, 50
    Hand-selected in RC, RNC and TOEP-S scenarios to create the scale imbalance that drives the paper's main effects.
  • Dependence strength rho = 0.5, 0.9
    Two hand-picked correlation levels in Toeplitz scenarios, chosen to represent mild and strong dependence.
  • Noise variance sigma_epsilon = derived so 90% of deviance is explained
    Set per scenario in Section 3 (Appendix A), making the signal-to-noise ratio a design choice rather than an estimated quantity.
  • Number and location of relevant covariates s = 10 (IND, UTOEP-S, TOEP-S), 15 (UTOEP-B)
    Hand-chosen in Section 3; block versus sparse placement changes the confusion phenomenon.
assumptions (4)
  • domain assumption The data generating process is a centered linear regression Y = X beta + epsilon with no intercept (Sections 1 and 3).
    All simulation scenarios and real-data analyses assume linearity; the guidelines do not cover nonlinear or non-additive structures.
  • domain assumption Covariates are Gaussian, X ~ N(0, Sigma), with diagonal or Toeplitz covariance (Section 3).
    The simulated dependence patterns are limited to these forms; heavy-tailed designs or non-Toeplitz correlations are not tested.
  • standard math The beta-min and n > s log(p) conditions from Buhlmann and Van De Geer (2011) guide the choice of simulation parameters (Section 3).
    These conditions are imported from the cited literature and assumed relevant to the qualitative behavior of the methods.
  • domain assumption Cross-validated glmnet with 10 folds and a grid of length 100 is a fair implementation basis for comparing all methods (Section 4).
    The comparison is only as fair as the default tuning implementations; no sensitivity analysis to tuning details is provided.

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Cite this review

Pith. "Pith review of Guidelines for LASSO and derivatives use under different dependence and scale structures." pith.science (2026). https://pith.science/paper/V4ZBKVSG

@misc{pith2026250608582,
  author       = {Pith},
  title        = {Pith review of: Guidelines for LASSO and derivatives use under different dependence and scale structures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4ZBKVSG}},
  note         = {Machine review of arXiv:2506.08582}
}
abstract

In a multivariate linear regression model with $p>1$ covariates, implementation of penalization techniques often implies a preliminary univariate standardization step. Although this prevents scale effects on the covariates selection procedure, possible dependence structures can be disrupted, leading to wrong results. This is particularly challenging in high-dimensional settings where $p \geq n$. In this paper, we analyze the standardization effect on the LASSO for different dependence-scales contexts by means of an extensive simulation study. Two distinct objectives are pursued: adequate covariate selection and proper predictive capability. Additionally, its behavior is compared with the one of some well-known or innovative competitors. This comparison is also extended to three real datasets facing different dependence-scales patterns. Eventually, we conclude with discussion and guidelines on the most suitable methodology for each case in terms of covariates selection or prediction.

Figures

Figures reproduced from arXiv: 2506.08582 by the authors.

Figure 1
Figure 1. Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for p = 100 selected in terms of Xr/Xus in scenarios IND (first row), RC.IND (second row) and RNC.IND (third row). The dashed line marks the s = 10 value. IND RC.IND RNC.IND RAW UNIV. RAW UNIV. RAW UNIV. METHOD MSE % Dev MSE % Dev MSE % Dev MSE % Dev MSE % Dev MSE % Dev (1.736) (0.9) (1.736) (0.9) (13.… view at source ↗
Figure 2
Figure 2. Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for p = 100 and ρ = 0.9 selected in terms of Xr/Xus in scenarios UTOEP-B (first row) and UTOEP-S (second row). The dashed line marks the s = 15 and s = 10 value for first and second row, respectively. UTOEP-B UTOEP-S RAW UNIV. RAW UNIV. METHOD MSE % Dev MSE % Dev MSE % Dev MSE % Dev (3.807) (0.9) (3.80… view at source ↗
Figure 3
Figure 3. Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for p = 100 and ρ = 0.9 selected in terms of Xr/Xus in scenarios RC.TOEP-S (first row) and RNC.TOEP-S (second row). The dashed line marks the s = 10 value. Eventually, we move to a more challenging framework: TOEP-S. We keep the Toeplitz de￾pendence structure of UTOEP-S but allow changes in the scale v… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Number of important covariates (dark [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for proposed algorithms taking p = 100 and Xr/Xus in scenarios UTOEP-B (the first row) and UTOEP-S (the second row) for ρ = 0.9 and n = 50. The dashed li…
Figure 6
Figure 6. Figure 6: Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for proposed algorithms taking p = 100 and selected in terms of Xr/Xus in scenarios RC.TOEP-S (the first row) and RNC.TOEP-S (the second row) for ρ = 0.9…
Figure 7
Figure 7. Figure 7: Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for proposed algorithms taking p = 100 and selected in terms of Xr/Xus in scenarios RC.IND (the first row) and RNC.IND (the second row) for n = 300. The …
Figure 8
Figure 8. Figure 8: Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for proposed algorithms taking p = 100 and Xr/Xus in scenarios UTOEP-B (the first row) and UTOEP-S (the second row) for ρ = 0.9 and n = 300. The dashed l…
Figure 9
Figure 9. Figure 9: Number of important covariates (dark left/right rectangular area) and noisy ones (soft left/right rectangular area) for proposed algorithms taking p = 100 and selected in terms of Xr/Xus in scenarios RC.TOEP-S (the first row) and RNC.TOEP-S (the second row) for ρ = 0.9…
Figure 10
Figure 10. Figure 10: Representative covariates relevance coefficients in increasing order in terms of R [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Number of selected covariates for the considered procedures for [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Number of selected covariates for the considered procedures for [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Number of selected covariates for the considered procedures for [PITH_FULL_IMAGE:figures/full_fig_p025_13.png]
Figure 14
Figure 14. Figure 14: Most adequate procedures to face dependence and/or scale effects to minimize the FDP [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]

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Pith tools

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