Asymptotic Theory for Graphical SLOPE: Precision Estimation and Pattern Convergence

Giovanni Bonaccolto; Ivan Hejn\'y; Jonas Wallin; Ma{\l}gorzata Bogdan; Philipp Kremer; Sandra Paterlini

arxiv: 2604.12771 · v1 · submitted 2026-04-14 · 🧮 math.ST · stat.AP· stat.ME· stat.ML· stat.TH

Asymptotic Theory for Graphical SLOPE: Precision Estimation and Pattern Convergence

Ivan Hejn\'y , Giovanni Bonaccolto , Philipp Kremer , Sandra Paterlini , Ma{\l}gorzata Bogdan , Jonas Wallin This is my paper

Pith reviewed 2026-05-10 14:06 UTC · model grok-4.3

classification 🧮 math.ST stat.APstat.MEstat.MLstat.TH

keywords graphical SLOPEprecision matrix estimationasymptotic theorypattern convergenceSLOPE penaltyelliptical distributionsmultivariate t-lossedge clustering

0 comments

The pith

The root-n scaled error of Graphical SLOPE converges to the unique minimizer of a strictly convex problem defined by the directional derivative of the SLOPE penalty, and the induced pattern converges.

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

This paper develops the asymptotic theory for Graphical SLOPE when estimating precision matrices in a fixed-dimensional regime where the number of variables stays fixed while the sample size grows. It establishes that the scaled estimation error converges in distribution to the solution of an optimization problem whose objective is built from the directional derivative of the SLOPE penalty. The same limit governs convergence of the SLOPE pattern, which encodes the grouping of edges into clusters of equal or similar strength. The authors also derive the limiting distributions when the data follow elliptical laws rather than Gaussian, showing that heavy tails inflate the variability of the Gaussian-loss version while the multivariate-t-loss version reduces that inflation. These results supply an asymptotic basis for understanding when the grouping property of SLOPE improves accuracy over plain GLASSO on matrices that contain repeated edge strengths.

Core claim

In the fixed-dimensional regime the root-n scaled estimation error of Graphical SLOPE converges to the unique minimizer of a strictly convex optimization problem defined through the directional derivative of the SLOPE penalty. Convergence of the induced SLOPE pattern is established at the same time, yielding an asymptotic description of the clustering structure that the estimator selects. Under elliptical distributions the limiting law for the Gaussian-loss estimator is obtained and the extra variability induced by heavy tails is quantified relative to the Gaussian benchmark. The limiting distribution for the t-loss estimator TSLOPE is likewise derived and shown to be advantageous under the

What carries the argument

The directional derivative of the SLOPE penalty, which defines the strictly convex limiting optimization problem that the scaled estimation error and the selected pattern both converge to.

Load-bearing premise

The dimension p is fixed while the sample size n tends to infinity and the data are generated from Gaussian or elliptical distributions.

What would settle it

In repeated simulations with fixed p and large n from Gaussian data, if the root-n errors fail to concentrate around the predicted minimizer or the selected edge clusters fail to stabilize at the predicted pattern, the convergence claims would be refuted.

Figures

Figures reproduced from arXiv: 2604.12771 by Giovanni Bonaccolto, Ivan Hejn\'y, Jonas Wallin, Ma{\l}gorzata Bogdan, Philipp Kremer, Sandra Paterlini.

**Figure 2.** Figure 2: Comparison of the asymptotic clustering error RMSE-CL (Equation ( [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** Figure 3: Comparison of the asymptotic GSLOPE and TSLOPE errors, defined as [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: The figure shows the oracle correlation matrix and the oracle precision matrix [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: The Figure shows from top left to bottom right: the oracle precision matrix, the [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: Boxplots of daily value-weighted returns for the 25 portfolios sorted by size and [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗

**Figure 7.** Figure 7: Time evolution of average daily value-weighted returns from January 3, 2000 to [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: Calinski-Harabasz index as a function of ln( [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗

**Figure 9.** Figure 9: Heatmap of cluster assignments for the off-diagonal entries of the TSLOPE es [PITH_FULL_IMAGE:figures/full_fig_p036_9.png] view at source ↗

**Figure 10.** Figure 10: Boxplots of the off-diagonal entries (upper-triangular part) of the yearly precision [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗

**Figure 11.** Figure 11: Network representation of the 25 portfolios based on the TSLOPE estimation at [PITH_FULL_IMAGE:figures/full_fig_p038_11.png] view at source ↗

read the original abstract

This paper studies Graphical SLOPE for precision matrix estimation, with emphasis on its ability to recover both sparsity and clusters of edges with equal or similar strength. In a fixed-dimensional regime, we establish that the root-$n$ scaled estimation error converges to the unique minimizer of a strictly convex optimization problem defined through the directional derivative of the SLOPE penalty. We also establish convergence of the induced SLOPE pattern, thereby obtaining an asymptotic characterization of the clustering structure selected by the estimator. A comparison with GLASSO shows that the grouping property of SLOPE can substantially improve estimation accuracy when the precision matrix exhibits structured edge patterns. To assess the effect of departures from Gaussianity, we then analyze Gaussian-loss precision matrix estimation under elliptical distributions. In this setting, we derive the limiting distribution and quantify the inflation in variability induced by heavy tails relative to the Gaussian benchmark. We also study TSLOPE, based on the multivariate $t$-loss, and derive its limiting distribution. The results show that TSLOPE offers clear advantages over GSLOPE under heavy-tailed data-generating mechanisms. Simulation evidence suggests that these qualitative conclusions persist in high-dimensional settings, and an empirical application shows that SLOPE-based estimators, especially TSLOPE, can uncover economically meaningful clustered dependence structures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Graphical SLOPE gets fixed-p asymptotics that characterize its pattern convergence and show gains over GLASSO for clustered edges, plus a TSLOPE extension that handles heavy tails.

read the letter

The paper establishes root-n consistency for Graphical SLOPE in fixed dimension, with the scaled error converging to the minimizer of a limiting convex program built from the loss Hessian and the directional derivative of the SLOPE penalty. It also proves that the induced pattern of equal off-diagonal entries converges, giving an asymptotic description of the clustering the estimator selects. The same framework is applied to elliptical distributions under Gaussian loss and to the multivariate-t loss, with explicit variance inflation factors for the heavy-tail case. TSLOPE comes out ahead of GSLOPE when tails are heavy. Simulations are said to carry the qualitative message into high dimensions, and an economic data example illustrates recovered clusters. These pieces are new relative to standard GLASSO asymptotics. The directional-derivative approach and the pattern-convergence result are the clearest additions. The comparison to GLASSO is direct and the robustness calculations are explicit. The fixed-p restriction is the main limit; most graphical-model work cares about p growing with n, so the results do not cover that regime. The abstract describes simulations only qualitatively, without numbers on the size of the accuracy gains or the exact designs used. Strict convexity and uniqueness of the limit are asserted but rest on steps that need checking in the full proofs. The overall argument follows standard convex M-estimator lines without circularity or self-fitting. This is for statisticians who work on penalized precision estimation and want theory for grouping penalties or robust losses. A reader who already knows GLASSO asymptotics will see the incremental value quickly. It deserves a serious referee because the pattern-convergence claim and the elliptical/TSLOPE extensions are concrete enough to review in detail. I would send it out.

Referee Report

2 major / 3 minor

Summary. The paper develops asymptotic theory for the Graphical SLOPE estimator for precision matrix estimation. In the fixed-dimensional regime (p fixed, n to infinity), the root-n scaled estimation error converges in distribution to the unique minimizer of a strictly convex limiting optimization problem whose objective combines the quadratic loss term with the directional derivative of the SLOPE penalty. The paper further establishes convergence of the induced SLOPE pattern, providing an asymptotic characterization of the clustering of edges with equal magnitudes. Extensions cover Gaussian-loss estimation under elliptical distributions (with explicit variance inflation) and the multivariate-t loss (TSLOPE). Comparisons with GLASSO, simulation studies, and an empirical application are included.

Significance. If the central claims hold, the work supplies a precise asymptotic description of both estimation error and pattern recovery for a grouped-sparsity penalty in graphical models. The pattern-convergence result is a genuine addition beyond standard sparsity analysis, and the explicit treatment of heavy-tailed elliptical and t-distributed data quantifies robustness gains. The derivation follows standard M-estimator arguments adapted to the directional derivative, which is a clean and reusable technique. These elements would be useful for researchers studying structured penalties and for practitioners selecting between SLOPE and lasso-type estimators when edge clusters are plausible.

major comments (2)

[Main theorem on root-n convergence] Main asymptotic theorem (fixed-p regime): the strict convexity of the limiting objective is asserted to guarantee uniqueness, but the argument relies on the directional derivative of the SLOPE penalty being strictly convex; an explicit verification or counter-example check for common choices of the penalty weights (e.g., when weights are not strictly decreasing) would strengthen the uniqueness claim.
[Pattern convergence result] Section on pattern convergence: the definition of the SLOPE pattern (clusters induced by equal-magnitude off-diagonal entries) is used to state convergence, yet the mapping from the limiting minimizer to the pattern is not shown to be continuous at the boundary points where ties occur; this continuity is load-bearing for the pattern-stability conclusion.

minor comments (3)

[Abstract and Introduction] The abstract and introduction use the term 'SLOPE pattern' without a self-contained definition; a brief inline definition or pointer to the precise mathematical object would improve readability.
[Simulation studies] Simulation section reports only qualitative agreement with the asymptotics; adding quantitative metrics (e.g., empirical coverage of the limiting distribution or pattern-recovery rates) would make the validation more convincing.
[Theoretical development] A few citations to the classical literature on directional derivatives for nonsmooth M-estimators (e.g., works on subdifferential calculus for convex penalties) are missing; adding them would place the technical contribution in clearer context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: Main asymptotic theorem (fixed-p regime): the strict convexity of the limiting objective is asserted to guarantee uniqueness, but the argument relies on the directional derivative of the SLOPE penalty being strictly convex; an explicit verification or counter-example check for common choices of the penalty weights (e.g., when weights are not strictly decreasing) would strengthen the uniqueness claim.

Authors: We agree that an explicit verification strengthens the uniqueness claim. The limiting objective is the sum of a strictly convex quadratic term (with positive definite Hessian given by the Kronecker product involving the covariance matrix) and the directional derivative of the convex SLOPE penalty. Strict convexity of the sum therefore follows from the quadratic term alone. In the revision we will add a short lemma verifying this under the standard assumption that the penalty weights are positive and non-increasing; the lemma will also note that non-strictly-decreasing weights do not destroy uniqueness because they cannot cancel the positive-definiteness of the quadratic Hessian. A brief counter-example check for the constant-weight (lasso) case will be included to illustrate the boundary. revision: yes
Referee: Section on pattern convergence: the definition of the SLOPE pattern (clusters induced by equal-magnitude off-diagonal entries) is used to state convergence, yet the mapping from the limiting minimizer to the pattern is not shown to be continuous at the boundary points where ties occur; this continuity is load-bearing for the pattern-stability conclusion.

Authors: We acknowledge that the pattern map is discontinuous precisely at points where two or more off-diagonal entries have equal magnitude. Because the limiting distribution of the root-n error is absolutely continuous (as the unique minimizer of a strictly convex objective driven by a non-degenerate Gaussian vector), the probability that the limiting minimizer lands exactly on a tie set is zero. In the revision we will insert a short argument establishing this measure-zero property and restate the pattern-convergence result as holding with probability approaching one, thereby removing the dependence on continuity at the boundary points. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on standard asymptotic M-estimator theory for convex penalized estimators in fixed dimension (p fixed, n→∞). The root-n scaled error is shown to converge to the unique minimizer of a limiting strictly convex program whose objective is the quadratic term from the loss Hessian plus the directional derivative of the SLOPE penalty; this limiting object is constructed from the data-generating distribution and the known properties of the penalty, not from quantities fitted to the same data. Pattern convergence follows directly from the uniqueness of the limiting minimizer. Extensions to elliptical distributions and the multivariate-t loss replace the score function while preserving the same directional-derivative argument. No step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the central claims are independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions for asymptotic statistics rather than new free parameters or invented entities.

axioms (2)

domain assumption Fixed dimension p with n to infinity
Enables root-n consistency and unique minimizer convergence in the stated regime.
domain assumption Data generated from Gaussian or elliptical distributions
Required for the limiting distributions of the Gaussian-loss and t-loss estimators.

pith-pipeline@v0.9.0 · 5555 in / 1375 out tokens · 46305 ms · 2026-05-10T14:06:10.304106+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

[1]

Wiley New York, 1958

Theodore Wilbur Anderson.An introduction to multivariate statistical analysis, vol- ume 2. Wiley New York, 1958

work page 1958
[2]

Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society: Series B (Methodological), 57(1):289–300, 1995

Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society: Series B (Methodological), 57(1):289–300, 1995

work page 1995
[3]

Identifying Network Hubs with the Partial Correlation Graphical LASSO

Ma lgorzata Bogdan, Adam Chojecki, Ivan Hejn´ y, Bartosz Ko lodziejek, and Jonas Wallin. Identifying network hubs with the partial correlation graphical lasso.arXiv preprint arXiv:2508.12258, 2025

work page internal anchor Pith review arXiv 2025
[4]

Cand` es

Ma lgorzata Bogdan, Ewout van den Berg, Chiara Sabatti, Weijie Su, and Emmanuel J. Cand` es. SLOPE—adaptive variable selection via convex optimization.The Annals of Applied Statistics, 9(3):1103–1140, 2015

work page 2015
[5]

Pattern recovery by slope.Applied and Computational Harmonic Analysis, 80:101810, 2026

Ma lgorzata Bogdan, Xavier Dupuis, Piotr Graczyk, Bartosz Ko lodziejek, Tomasz Skalski, Patrick Tardivel, and Maciej Wilczy´ nski. Pattern recovery by slope.Applied and Computational Harmonic Analysis, 80:101810, 2026

work page 2026
[6]

A constrainedℓ 1 minimization approach to sparse precision matrix estimation.Journal of the American Statistical Association, 106(494):594–607, 2011

Tony Cai, Weidong Liu, and Xi Luo. A constrainedℓ 1 minimization approach to sparse precision matrix estimation.Journal of the American Statistical Association, 106(494):594–607, 2011

work page 2011
[7]

Calinski and J

T. Calinski and J. Harabasz. A dendrite method for cluster analysis.Communications in Statistics - Theory and Methods, 3(1):1–27, 1974

work page 1974
[8]

R. Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2):223–236, 2001

work page 2001
[9]

Fama and Kenneth R

Eugene F. Fama and Kenneth R. French. Common risk factors in the returns on stocks and bonds.Journal of Financial Economics, 33(1):3–56, 1993

work page 1993
[10]

Fama and Kenneth R

Eugene F. Fama and Kenneth R. French. A five-factor asset pricing model.Journal of Financial Economics, 116(1):1–22, 2015

work page 2015
[11]

Sparse inverse covariance estimation with the graphical lasso.Biostatistics, 9(3):432–441, 2008

Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso.Biostatistics, 9(3):432–441, 2008

work page 2008
[12]

Asymptotics for lasso-type estimators.Annals of Statistics, 28(5):1356–1378, 2000

Wenjiang Fu and Keith Knight. Asymptotics for lasso-type estimators.Annals of Statistics, 28(5):1356–1378, 2000

work page 2000
[13]

From graphical lasso to atomic norms: High- dimensional pattern recovery.arXiv preprint arXiv:2506.13353, 2025

Piotr Graczyk, Hideto Nakashima, et al. From graphical lasso to atomic norms: High- dimensional pattern recovery.arXiv preprint arXiv:2506.13353, 2025

work page arXiv 2025
[14]

Asymptotic distribution of low- dimensional patterns induced by non-differentiable regularizers under general loss func- tions.arXiv preprint arXiv:2506.12621, 2025

Ivan Hejn´ y, Jonas Wallin, and Ma lgorzata Bogdan. Asymptotic distribution of low- dimensional patterns induced by non-differentiable regularizers under general loss func- tions.arXiv preprint arXiv:2506.12621, 2025. 25

work page arXiv 2025
[15]

Unveiling low- dimensional patterns induced by convex non-differentiable regularizers.Annals of the Institute of Statistical Mathematics, pages 1–36, 2025

Ivan Hejn´ y, Jonas Wallin, Ma lgorzata Bogdan, and Micha l Kos. Unveiling low- dimensional patterns induced by convex non-differentiable regularizers.Annals of the Institute of Statistical Mathematics, pages 1–36, 2025

work page 2025
[16]

Lauritzen

Søren Højsgaard and Steffen L. Lauritzen. Graphical gaussian models with edge and vertex symmetries.Journal of the Royal Statistical Society: Series B, 70(5):1005–1027, 2008

work page 2008
[17]

Kremer, Sangkyun Lee, Ma lgorzata Bogdan, and Sandra Paterlini

Philipp J. Kremer, Sangkyun Lee, Ma lgorzata Bogdan, and Sandra Paterlini. Sparse portfolio selection via the sortedℓ1-norm.Journal of Banking and Finance, 110, 2020

work page 2020
[18]

Clarendon Press, 1996

Steffen L Lauritzen.Graphical models, volume 17. Clarendon Press, 1996

work page 1996
[19]

High-dimensional graphs and variable selec- tion with the Lasso.The Annals of Statistics, 34(3):1436–1462, 2006

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

work page 2006
[20]

Wainwright, Garvesh Raskutti, and Bin Yu

Pradeep Ravikumar, Martin J. Wainwright, Garvesh Raskutti, and Bin Yu. High- dimensional covariance estimation by minimizingℓ 1-penalized log-determinant diver- gence.Electronic Journal of Statistics, 5:935–980, 2011

work page 2011
[21]

Kremer, Piotr Sobczyk, Ma lgorzata Bogdan, and Sandra Paterlini

Riccardo Riccobello, Giovanni Bonaccolto, Philipp J. Kremer, Piotr Sobczyk, Ma lgorzata Bogdan, and Sandra Paterlini. Sparse graphical modelling for global mini- mum variance portfolio.Computational Management Science, 22(2), 2025

work page 2025
[22]

Stephen A. Ross. The arbitrage theory of capital asset pricing.Journal of Economic Theory, 13(3):341–360, 1976

work page 1976
[23]

PhD thesis, Wroc law University of Science and Technology, Wroc law, Poland, 2019

Piotr Sobczyk.Identifying Low-Dimensional Structures Through Model Selection in High-Dimensional Data. PhD thesis, Wroc law University of Science and Technology, Wroc law, Poland, 2019

work page 2019
[24]

A. W. van der Vaart.Asymptotic Statistics. Cambridge Series in Statistical and Prob- abilistic Mathematics. Cambridge University Press, Cambridge, 1998. 26 7 Appendix 7.1 Pattern, Pattern Space and Clustering Error of SLOPE Example 7.1.We illustrate the notion of a SLOPE pattern, the pattern space on the following example. Consider a precision matrix Θ 0 ...

work page 1998
[25]

Consequently, |ℓ(X,Θ)| ≤ 1 2 |log det(Θ)|+ ν+p 2 |log(ν+X T ΘX)| ≤ p 2 max(|logr −|,|logr +|) + ν+p 2 max |log(ν)|,log(ν+r +∥X∥2 2)

The term log(ν+X T ΘX) is bounded from below by log(ν) and from above by log(ν+r +∥X∥2 2). Consequently, |ℓ(X,Θ)| ≤ 1 2 |log det(Θ)|+ ν+p 2 |log(ν+X T ΘX)| ≤ p 2 max(|logr −|,|logr +|) + ν+p 2 max |log(ν)|,log(ν+r +∥X∥2 2) . This provides an integrable envelope. Specifically, for an elliptical vectorX= Θ −1/2 0 uR, the expectationE[log(ν+r +∥X∥2 2)] behav...

work page 2000

[1] [1]

Wiley New York, 1958

Theodore Wilbur Anderson.An introduction to multivariate statistical analysis, vol- ume 2. Wiley New York, 1958

work page 1958

[2] [2]

Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society: Series B (Methodological), 57(1):289–300, 1995

Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society: Series B (Methodological), 57(1):289–300, 1995

work page 1995

[3] [3]

Identifying Network Hubs with the Partial Correlation Graphical LASSO

Ma lgorzata Bogdan, Adam Chojecki, Ivan Hejn´ y, Bartosz Ko lodziejek, and Jonas Wallin. Identifying network hubs with the partial correlation graphical lasso.arXiv preprint arXiv:2508.12258, 2025

work page internal anchor Pith review arXiv 2025

[4] [4]

Cand` es

Ma lgorzata Bogdan, Ewout van den Berg, Chiara Sabatti, Weijie Su, and Emmanuel J. Cand` es. SLOPE—adaptive variable selection via convex optimization.The Annals of Applied Statistics, 9(3):1103–1140, 2015

work page 2015

[5] [5]

Pattern recovery by slope.Applied and Computational Harmonic Analysis, 80:101810, 2026

Ma lgorzata Bogdan, Xavier Dupuis, Piotr Graczyk, Bartosz Ko lodziejek, Tomasz Skalski, Patrick Tardivel, and Maciej Wilczy´ nski. Pattern recovery by slope.Applied and Computational Harmonic Analysis, 80:101810, 2026

work page 2026

[6] [6]

A constrainedℓ 1 minimization approach to sparse precision matrix estimation.Journal of the American Statistical Association, 106(494):594–607, 2011

Tony Cai, Weidong Liu, and Xi Luo. A constrainedℓ 1 minimization approach to sparse precision matrix estimation.Journal of the American Statistical Association, 106(494):594–607, 2011

work page 2011

[7] [7]

Calinski and J

T. Calinski and J. Harabasz. A dendrite method for cluster analysis.Communications in Statistics - Theory and Methods, 3(1):1–27, 1974

work page 1974

[8] [8]

R. Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2):223–236, 2001

work page 2001

[9] [9]

Fama and Kenneth R

Eugene F. Fama and Kenneth R. French. Common risk factors in the returns on stocks and bonds.Journal of Financial Economics, 33(1):3–56, 1993

work page 1993

[10] [10]

Fama and Kenneth R

Eugene F. Fama and Kenneth R. French. A five-factor asset pricing model.Journal of Financial Economics, 116(1):1–22, 2015

work page 2015

[11] [11]

Sparse inverse covariance estimation with the graphical lasso.Biostatistics, 9(3):432–441, 2008

Jerome Friedman, Trevor Hastie, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso.Biostatistics, 9(3):432–441, 2008

work page 2008

[12] [12]

Asymptotics for lasso-type estimators.Annals of Statistics, 28(5):1356–1378, 2000

Wenjiang Fu and Keith Knight. Asymptotics for lasso-type estimators.Annals of Statistics, 28(5):1356–1378, 2000

work page 2000

[13] [13]

From graphical lasso to atomic norms: High- dimensional pattern recovery.arXiv preprint arXiv:2506.13353, 2025

Piotr Graczyk, Hideto Nakashima, et al. From graphical lasso to atomic norms: High- dimensional pattern recovery.arXiv preprint arXiv:2506.13353, 2025

work page arXiv 2025

[14] [14]

Asymptotic distribution of low- dimensional patterns induced by non-differentiable regularizers under general loss func- tions.arXiv preprint arXiv:2506.12621, 2025

Ivan Hejn´ y, Jonas Wallin, and Ma lgorzata Bogdan. Asymptotic distribution of low- dimensional patterns induced by non-differentiable regularizers under general loss func- tions.arXiv preprint arXiv:2506.12621, 2025. 25

work page arXiv 2025

[15] [15]

Unveiling low- dimensional patterns induced by convex non-differentiable regularizers.Annals of the Institute of Statistical Mathematics, pages 1–36, 2025

Ivan Hejn´ y, Jonas Wallin, Ma lgorzata Bogdan, and Micha l Kos. Unveiling low- dimensional patterns induced by convex non-differentiable regularizers.Annals of the Institute of Statistical Mathematics, pages 1–36, 2025

work page 2025

[16] [16]

Lauritzen

Søren Højsgaard and Steffen L. Lauritzen. Graphical gaussian models with edge and vertex symmetries.Journal of the Royal Statistical Society: Series B, 70(5):1005–1027, 2008

work page 2008

[17] [17]

Kremer, Sangkyun Lee, Ma lgorzata Bogdan, and Sandra Paterlini

Philipp J. Kremer, Sangkyun Lee, Ma lgorzata Bogdan, and Sandra Paterlini. Sparse portfolio selection via the sortedℓ1-norm.Journal of Banking and Finance, 110, 2020

work page 2020

[18] [18]

Clarendon Press, 1996

Steffen L Lauritzen.Graphical models, volume 17. Clarendon Press, 1996

work page 1996

[19] [19]

High-dimensional graphs and variable selec- tion with the Lasso.The Annals of Statistics, 34(3):1436–1462, 2006

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

work page 2006

[20] [20]

Wainwright, Garvesh Raskutti, and Bin Yu

Pradeep Ravikumar, Martin J. Wainwright, Garvesh Raskutti, and Bin Yu. High- dimensional covariance estimation by minimizingℓ 1-penalized log-determinant diver- gence.Electronic Journal of Statistics, 5:935–980, 2011

work page 2011

[21] [21]

Kremer, Piotr Sobczyk, Ma lgorzata Bogdan, and Sandra Paterlini

Riccardo Riccobello, Giovanni Bonaccolto, Philipp J. Kremer, Piotr Sobczyk, Ma lgorzata Bogdan, and Sandra Paterlini. Sparse graphical modelling for global mini- mum variance portfolio.Computational Management Science, 22(2), 2025

work page 2025

[22] [22]

Stephen A. Ross. The arbitrage theory of capital asset pricing.Journal of Economic Theory, 13(3):341–360, 1976

work page 1976

[23] [23]

PhD thesis, Wroc law University of Science and Technology, Wroc law, Poland, 2019

Piotr Sobczyk.Identifying Low-Dimensional Structures Through Model Selection in High-Dimensional Data. PhD thesis, Wroc law University of Science and Technology, Wroc law, Poland, 2019

work page 2019

[24] [24]

A. W. van der Vaart.Asymptotic Statistics. Cambridge Series in Statistical and Prob- abilistic Mathematics. Cambridge University Press, Cambridge, 1998. 26 7 Appendix 7.1 Pattern, Pattern Space and Clustering Error of SLOPE Example 7.1.We illustrate the notion of a SLOPE pattern, the pattern space on the following example. Consider a precision matrix Θ 0 ...

work page 1998

[25] [25]

Consequently, |ℓ(X,Θ)| ≤ 1 2 |log det(Θ)|+ ν+p 2 |log(ν+X T ΘX)| ≤ p 2 max(|logr −|,|logr +|) + ν+p 2 max |log(ν)|,log(ν+r +∥X∥2 2)

The term log(ν+X T ΘX) is bounded from below by log(ν) and from above by log(ν+r +∥X∥2 2). Consequently, |ℓ(X,Θ)| ≤ 1 2 |log det(Θ)|+ ν+p 2 |log(ν+X T ΘX)| ≤ p 2 max(|logr −|,|logr +|) + ν+p 2 max |log(ν)|,log(ν+r +∥X∥2 2) . This provides an integrable envelope. Specifically, for an elliptical vectorX= Θ −1/2 0 uR, the expectationE[log(ν+r +∥X∥2 2)] behav...

work page 2000