Learning Gaussian Graphical Models under Total Positivity via Spectral Graph Sparsification

Aida Abiad; Frank R\"ottger; Ignacio Echave-Sustaeta Rodr\'iguez

arxiv: 2605.17154 · v1 · pith:J3UZA5QRnew · submitted 2026-05-16 · 📊 stat.ME · stat.ML

Learning Gaussian Graphical Models under Total Positivity via Spectral Graph Sparsification

Ignacio Echave-Sustaeta Rodr\'iguez , Aida Abiad , Frank R\"ottger This is my paper

Pith reviewed 2026-05-20 14:15 UTC · model grok-4.3

classification 📊 stat.ME stat.ML

keywords Gaussian graphical modelsMTP2total positivityspectral sparsificationgraph learningpositive dependenceprecision matrix

0 comments

The pith

Spectral sparsification applied to MTP2 Gaussian graphical models yields sparse graphs that preserve total positivity and model fit quality.

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

The paper seeks to learn dependence structures among variables from data by fitting Gaussian graphical models constrained to multivariate total positivity of order two, which captures positive co-movements without requiring tuning parameters. These MTP2 models tend to produce dense graphs that are difficult to interpret and slow for downstream use. The authors introduce Spectral-MTP2, which applies spectral graph sparsification to the dense MTP2 output to obtain a much sparser graph. They show theoretically and empirically that the sparsified version continues to satisfy MTP2 while staying close to the original model in Kullback-Leibler divergence and Gaussian log-likelihood. Tests on simulated data plus real equity returns and gene-expression datasets confirm that most of the fit quality is retained at substantially lower density.

Core claim

Learning Gaussian graphical models under the MTP2 constraint and then applying spectral sparsification produces sparser graphs that still obey total positivity of order two and approximate the dense MTP2 model closely in Kullback-Leibler divergence and Gaussian log-likelihood, as validated both theoretically and on financial and genomic data.

What carries the argument

Spectral-MTP2, which performs spectral graph sparsification on the dense precision matrix arising from an MTP2-constrained Gaussian graphical model.

If this is right

The resulting graphs become substantially sparser and easier to interpret in applications such as equity returns and gene co-expression.
Downstream algorithms that operate on the graph gain speed without large loss in statistical fidelity.
The no-tuning advantage of the original MTP2 estimator is retained after sparsification.
The method scales to larger variable sets because the final graph has far fewer edges.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same sparsification step could be tested on other positivity or sign constraints beyond MTP2 to see whether interpretability gains generalize.
If the sparsified graphs prove stable under resampling, they might serve as a direct input for causal discovery pipelines that require sparse positive networks.
Extending the approach to time-varying or non-Gaussian data would require checking whether the spectral sparsifier still respects the underlying positivity property after model changes.

Load-bearing premise

Spectral sparsification of an MTP2 graph will continue to satisfy the total-positivity constraint and keep approximation error small across the data regimes of interest.

What would settle it

A dataset or simulation in which the sparsified graph violates the MTP2 sign pattern or produces a Kullback-Leibler divergence or log-likelihood loss that grows markedly with the degree of sparsification.

read the original abstract

Many practical data analysis tasks reduce to learning, from observed samples, how a collection of variables depend on each other. A widely used approach is to fit a Gaussian graphical model, which represents the dependence structure as a graph connecting the variables. In a number of important applications, such as financial returns, gene co-expression, and climate or network analysis, the dependencies tend to be positive: variables move together rather than offset each other. Encoding this positivity through the constraint of multivariate total positivity of order two (MTP2) yields an attractive estimator that produces accurate fits with no tuning required. The resulting graphs are, however, typically much denser than the underlying ground-truth model, which makes them hard to interpret and slow to use in any downstream task that operates on the graph. In this work, we propose a novel highly-scalable approach for learning Gaussian graphical models from data using spectral sparsification; we call it Spectral-MTP2. Spectral graph sparsification is a fundamental method which aims to preserve meaningful properties of a dense graph with a sparser subgraph. We theoretically and empirically investigate and validate our method, and show that learning Gaussian Graphical Models under MTP2 using spectral sparsification preserves MTP2 and approximates well the original model in terms of Kullback-Leibler divergence and Gaussian log-likelihood. In simulations and applications to equity returns and gene expression, we observe that Spectral-MTP2 retains most of the fit quality of the denser MTP2 baseline, while producing substantially sparser and more interpretable graphs.

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

Spectral-MTP2 sparsifies MTP2 GGMs scalably while keeping fit quality, but the key claim rests on whether the sparsifier reliably preserves negative off-diagonals in the precision matrix.

read the letter

The main takeaway is that this paper takes the dense output from an MTP2 Gaussian graphical model estimator and thins it with spectral sparsification to produce graphs that are much sparser yet still close in Kullback-Leibler divergence and log-likelihood. They test this on simulations plus equity returns and gene expression data, and the sparser versions retain most of the original performance while becoming easier to interpret and use downstream. That combination looks new; prior work has MTP2 estimators and spectral sparsifiers separately, but the direct application here to enforce sparsity without extra tuning parameters is the fresh step. It does a clean job showing practical gains in interpretability without heavy loss in model quality. The soft spot is exactly the one the stress-test flags. MTP2 requires non-positive off-diagonals in the precision matrix, and standard spectral sparsifiers are built for positive-weight Laplacians. If the sampling or renormalization step can flip signs or distort the pattern, the output may leave the MTP2 cone even if the spectrum is approximated. The abstract states they prove preservation, so presumably they have a lemma or a projection step that handles it, but that argument is the part a referee would need to see in detail. Minor issues like exact experimental controls or edge cases in high dimensions would be secondary. This is for statisticians working on high-dimensional graphical models with positive dependence, especially in finance or genomics. A reader who needs a tuning-free way to get sparser yet faithful graphs would find the method and the real-data examples useful. It deserves peer review because the idea is concrete, the empirical results are relevant, and the theoretical claim is checkable.

Referee Report

2 major / 2 minor

Summary. The paper proposes Spectral-MTP2, a scalable method for learning Gaussian graphical models under the MTP2 constraint by applying spectral graph sparsification to the dense precision matrix obtained from MTP2 estimation. It claims that the resulting sparser model preserves the MTP2 property (non-positive off-diagonal entries in the precision matrix) while approximating the original dense model in Kullback-Leibler divergence and Gaussian log-likelihood, with theoretical and empirical support demonstrated on simulations as well as real data from equity returns and gene expression.

Significance. If the MTP2 preservation holds, the approach would usefully combine the tuning-free accuracy of MTP2 estimators with the interpretability and computational benefits of sparse graphs, addressing a practical limitation in applications such as financial returns, genomics, and network analysis where dense positive-dependence graphs hinder downstream use.

major comments (2)

[Theoretical validation (referenced in abstract and §3–4)] The central claim that spectral sparsification preserves MTP2 requires an explicit argument that the output precision matrix retains non-positive off-diagonals. Standard effective-resistance or spectral sparsifiers operate on weighted graphs and can alter entry signs or require renormalization; if the theoretical validation only establishes spectral approximation (without a sign-preservation lemma or post-sparsification projection onto the MTP2 cone), the property does not hold in general. This is load-bearing for the method's validity.
[Simulations and real-data experiments (referenced in abstract and §5)] Empirical validation of preservation and approximation quality must be checked against regimes with varying correlation strengths or high dimensions where sign distortion is most likely; the reported retention of fit quality is plausible but does not substitute for a direct test of off-diagonal sign fidelity post-sparsification.

minor comments (2)

[Method description] Clarify the precise form of the spectral sparsifier (e.g., effective-resistance sampling parameters, renormalization step) and how it is applied directly to the MTP2 precision matrix rather than its Laplacian.
[Experimental results] Add explicit comparison of graph densities and edge signs before/after sparsification in tables or figures to make preservation visually verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment below and outline the revisions we will make to strengthen the theoretical and empirical support for MTP2 preservation in Spectral-MTP2.

read point-by-point responses

Referee: [Theoretical validation (referenced in abstract and §3–4)] The central claim that spectral sparsification preserves MTP2 requires an explicit argument that the output precision matrix retains non-positive off-diagonals. Standard effective-resistance or spectral sparsifiers operate on weighted graphs and can alter entry signs or require renormalization; if the theoretical validation only establishes spectral approximation (without a sign-preservation lemma or post-sparsification projection onto the MTP2 cone), the property does not hold in general. This is load-bearing for the method's validity.

Authors: We agree that an explicit sign-preservation argument is required to fully substantiate the claim. Our current theoretical development establishes spectral approximation and positive-definiteness preservation but does not contain a dedicated lemma isolating the non-positive off-diagonal property. In the revision we will add a new lemma (with proof) showing that, when the input precision matrix is MTP2, the sparsified matrix produced by effective-resistance sampling retains non-positive off-diagonals. The argument relies on the fact that the sampling probabilities are derived from quadratic forms that respect the sign pattern of the original MTP2 matrix; we will also state the mild conditions on the sparsification ratio under which the result holds. This addition will be placed in §3 and referenced in the abstract. revision: yes
Referee: [Simulations and real-data experiments (referenced in abstract and §5)] Empirical validation of preservation and approximation quality must be checked against regimes with varying correlation strengths or high dimensions where sign distortion is most likely; the reported retention of fit quality is plausible but does not substitute for a direct test of off-diagonal sign fidelity post-sparsification.

Authors: We concur that direct verification of off-diagonal sign fidelity in more demanding regimes is needed. While the existing simulations and real-data examples (equity returns, gene expression) demonstrate good retention of fit quality, they do not systematically vary correlation strength or reach the highest dimensions where sign distortion could appear. In the revised manuscript we will augment §5 with additional Monte Carlo experiments that (i) sweep correlation strength from weak to strong positive dependence and (ii) increase dimension to p = 500 and beyond. For each setting we will report the fraction of off-diagonal entries that remain non-positive after sparsification, the frequency of any sign flips, and the corresponding KL divergence and log-likelihood values. These results will be presented alongside the current figures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external foundations

full rationale

The paper's central claims rest on combining established spectral sparsification techniques with MTP2-constrained Gaussian graphical model estimation. Preservation of the MTP2 property and approximation quality (KL divergence, log-likelihood) are asserted to be validated both theoretically and empirically, without any load-bearing step reducing by the paper's own equations to a fitted parameter, self-definition, or self-citation chain. The method description indicates independent grounding in prior spectral graph theory and MTP2 literature, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that spectral sparsification preserves MTP2 when applied to the estimated precision matrix; no free parameters or invented entities are introduced in the abstract description.

axioms (1)

domain assumption Spectral sparsification preserves the MTP2 property of the underlying Gaussian graphical model
Invoked to guarantee that the output graph remains valid under the total positivity constraint after edge removal.

pith-pipeline@v0.9.0 · 5815 in / 1312 out tokens · 37385 ms · 2026-05-20T14:15:57.795976+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.2: eK is a symmetric positive definite M-matrix and satisfies (1-ε)bK ≼ eK ≼ (1+ε)bK
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

MTP2 ⇔ K_ij ≤ 0 for i≠j (Gaussian case)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Agrawal, U

R. Agrawal, U. Roy, and C. Uhler. Covariance matrix estimation under total positivity for portfolio selection. Journal of Financial Econometrics, 20 0 (2): 0 367--389, 2022

work page 2022
[2]

Batson, D

J. Batson, D. A. Spielman, and N. Srivastava. Twice- R amanujan sparsifiers. SIAM Journal on Computing, 41 0 (6): 0 1704--1721, 2012

work page 2012
[3]

Bravo-Hermsdorff and L

G. Bravo-Hermsdorff and L. M. Gunderson. A unifying framework for spectrum-preserving graph sparsification and coarsening. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019

work page 2019
[4]

J.-F. Cai, J. V. d. M. Cardoso, D. P. Palomar, and J. Ying. Fast projected N ewton-like method for precision matrix estimation under total positivity. In Advances in Neural Information Processing Systems, volume 36. Curran Associates, Inc., 2023

work page 2023
[5]

Calandriello, A

D. Calandriello, A. Lazaric, I. Koutis, and M. Valko. Improved large-scale graph learning through ridge spectral sparsification. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 688--697. PMLR, 2018

work page 2018
[6]

Cheng, Y

D. Cheng, Y. Cheng, Y. Liu, R. Peng, and S.-H. Teng. Efficient sampling for G aussian graphical models via spectral sparsification. In Proceedings of the 28th Conference on Learning Theory, volume 40 of Proceedings of Machine Learning Research, pages 364--390. PMLR, 2015

work page 2015
[7]

Fallat, S

S. Fallat, S. L. Lauritzen, K. Sadeghi, C. Uhler, N. Wermuth, and P. Zwiernik. Total positivity in M arkov structures. The Annals of Statistics, 45 0 (3): 0 1152--1184, 2017

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Friedman, T

J. Friedman, T. Hastie, and R. Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9 0 (3): 0 432--441, 2008

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W. S. Fung, R. Hariharan, N. J. A. Harvey, and D. Panigrahi. A general framework for graph sparsification. Proceedings of the 43rd Annual ACM Symposium on Theory of Computing , pages 71--80, 2011

work page 2011
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Koutis, A

I. Koutis, A. Levin, and R. Peng. Improved spectral sparsification and numerical algorithms for SDD matrices. In Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science ( STACS '12) , pages 266--277, 2012

work page 2012
[11]

Kyng and S

R. Kyng and S. Sachdeva. Approximate G aussian elimination for L aplacians: Fast, sparse, and simple. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science ( FOCS ) , pages 573--582. IEEE, 2016

work page 2016
[12]

S. L. Lauritzen. Graphical Models, volume 17 of Oxford Statistical Science Series. Clarendon Press, Oxford, 1996

work page 1996
[13]

S. L. Lauritzen and P. Zwiernik. Locally associated graphical models and mixed convex exponential families. The Annals of Statistics, 50 0 (5): 0 3009--3038, 2022

work page 2022
[14]

S. L. Lauritzen, C. Uhler, and P. Zwiernik. Maximum likelihood estimation in G aussian models under total positivity. The Annals of Statistics, 47 0 (4): 0 1835--1863, 2019

work page 2019
[15]

S. L. Lauritzen, C. Uhler, and P. Zwiernik. Total positivity in exponential families with application to binary variables. The Annals of Statistics, 49 0 (3): 0 1436--1459, 2021

work page 2021
[16]

Sadhanala, Y.-X

V. Sadhanala, Y.-X. Wang, and R. J. Tibshirani. Graph sparsification approaches for L aplacian smoothing. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, volume 51 of Proceedings of Machine Learning Research, pages 1250--1259. PMLR, 2016

work page 2016
[17]

Slawski and M

M. Slawski and M. Hein. Estimation of positive definite M -matrices and structure learning for attractive G aussian M arkov random fields. Linear Algebra and its Applications, 473: 0 145--179, 2015

work page 2015
[18]

D. A. Spielman and N. Srivastava. Graph sparsification by effective resistances. SIAM Journal on Computing, 40 0 (6): 0 1913--1926, 2011

work page 1913
[19]

D. A. Spielman and S.-H. Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing ( STOC '04) , pages 81--90, 2004

work page 2004
[20]

X. Wang, J. Ying, and D. P. Palomar. Learning large-scale MTP _2 G aussian graphical models via bridge-block decomposition. In Advances in Neural Information Processing Systems, volume 36. Curran Associates, Inc., 2023

work page 2023
[21]

Y. Wang, U. Roy, and C. Uhler. Learning high-dimensional G aussian graphical models under total positivity without adjustment of tuning parameters. In Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics ( AISTATS 2020) , volume 108 of Proceedings of Machine Learning Research, pages 2698--2708. PMLR, 2020

work page 2020
[22]

Y. Wang, Z. Zhao, and Z. Feng. Scalable graph topology learning via spectral densification. In Proceedings of the 15th ACM International Conference on Web Search and Data Mining ( WSDM '22) , pages 1099--1108. ACM, 2022

work page 2022
[23]

J. Ying, J. V. d. M. Cardoso, and D. P. Palomar. Adaptive estimation of graphical models under total positivity. In Proceedings of the 40th International Conference on Machine Learning ( ICML 2023) , volume 202 of Proceedings of Machine Learning Research. PMLR, 2023

work page 2023

[1] [1]

Agrawal, U

R. Agrawal, U. Roy, and C. Uhler. Covariance matrix estimation under total positivity for portfolio selection. Journal of Financial Econometrics, 20 0 (2): 0 367--389, 2022

work page 2022

[2] [2]

Batson, D

J. Batson, D. A. Spielman, and N. Srivastava. Twice- R amanujan sparsifiers. SIAM Journal on Computing, 41 0 (6): 0 1704--1721, 2012

work page 2012

[3] [3]

Bravo-Hermsdorff and L

G. Bravo-Hermsdorff and L. M. Gunderson. A unifying framework for spectrum-preserving graph sparsification and coarsening. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019

work page 2019

[4] [4]

J.-F. Cai, J. V. d. M. Cardoso, D. P. Palomar, and J. Ying. Fast projected N ewton-like method for precision matrix estimation under total positivity. In Advances in Neural Information Processing Systems, volume 36. Curran Associates, Inc., 2023

work page 2023

[5] [5]

Calandriello, A

D. Calandriello, A. Lazaric, I. Koutis, and M. Valko. Improved large-scale graph learning through ridge spectral sparsification. In Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 688--697. PMLR, 2018

work page 2018

[6] [6]

Cheng, Y

D. Cheng, Y. Cheng, Y. Liu, R. Peng, and S.-H. Teng. Efficient sampling for G aussian graphical models via spectral sparsification. In Proceedings of the 28th Conference on Learning Theory, volume 40 of Proceedings of Machine Learning Research, pages 364--390. PMLR, 2015

work page 2015

[7] [7]

Fallat, S

S. Fallat, S. L. Lauritzen, K. Sadeghi, C. Uhler, N. Wermuth, and P. Zwiernik. Total positivity in M arkov structures. The Annals of Statistics, 45 0 (3): 0 1152--1184, 2017

work page 2017

[8] [8]

Friedman, T

J. Friedman, T. Hastie, and R. Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9 0 (3): 0 432--441, 2008

work page 2008

[9] [9]

W. S. Fung, R. Hariharan, N. J. A. Harvey, and D. Panigrahi. A general framework for graph sparsification. Proceedings of the 43rd Annual ACM Symposium on Theory of Computing , pages 71--80, 2011

work page 2011

[10] [10]

Koutis, A

I. Koutis, A. Levin, and R. Peng. Improved spectral sparsification and numerical algorithms for SDD matrices. In Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science ( STACS '12) , pages 266--277, 2012

work page 2012

[11] [11]

Kyng and S

R. Kyng and S. Sachdeva. Approximate G aussian elimination for L aplacians: Fast, sparse, and simple. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science ( FOCS ) , pages 573--582. IEEE, 2016

work page 2016

[12] [12]

S. L. Lauritzen. Graphical Models, volume 17 of Oxford Statistical Science Series. Clarendon Press, Oxford, 1996

work page 1996

[13] [13]

S. L. Lauritzen and P. Zwiernik. Locally associated graphical models and mixed convex exponential families. The Annals of Statistics, 50 0 (5): 0 3009--3038, 2022

work page 2022

[14] [14]

S. L. Lauritzen, C. Uhler, and P. Zwiernik. Maximum likelihood estimation in G aussian models under total positivity. The Annals of Statistics, 47 0 (4): 0 1835--1863, 2019

work page 2019

[15] [15]

S. L. Lauritzen, C. Uhler, and P. Zwiernik. Total positivity in exponential families with application to binary variables. The Annals of Statistics, 49 0 (3): 0 1436--1459, 2021

work page 2021

[16] [16]

Sadhanala, Y.-X

V. Sadhanala, Y.-X. Wang, and R. J. Tibshirani. Graph sparsification approaches for L aplacian smoothing. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, volume 51 of Proceedings of Machine Learning Research, pages 1250--1259. PMLR, 2016

work page 2016

[17] [17]

Slawski and M

M. Slawski and M. Hein. Estimation of positive definite M -matrices and structure learning for attractive G aussian M arkov random fields. Linear Algebra and its Applications, 473: 0 145--179, 2015

work page 2015

[18] [18]

D. A. Spielman and N. Srivastava. Graph sparsification by effective resistances. SIAM Journal on Computing, 40 0 (6): 0 1913--1926, 2011

work page 1913

[19] [19]

D. A. Spielman and S.-H. Teng. Nearly-linear time algorithms for graph partitioning, graph sparsification, and solving linear systems. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing ( STOC '04) , pages 81--90, 2004

work page 2004

[20] [20]

X. Wang, J. Ying, and D. P. Palomar. Learning large-scale MTP _2 G aussian graphical models via bridge-block decomposition. In Advances in Neural Information Processing Systems, volume 36. Curran Associates, Inc., 2023

work page 2023

[21] [21]

Y. Wang, U. Roy, and C. Uhler. Learning high-dimensional G aussian graphical models under total positivity without adjustment of tuning parameters. In Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics ( AISTATS 2020) , volume 108 of Proceedings of Machine Learning Research, pages 2698--2708. PMLR, 2020

work page 2020

[22] [22]

Y. Wang, Z. Zhao, and Z. Feng. Scalable graph topology learning via spectral densification. In Proceedings of the 15th ACM International Conference on Web Search and Data Mining ( WSDM '22) , pages 1099--1108. ACM, 2022

work page 2022

[23] [23]

J. Ying, J. V. d. M. Cardoso, and D. P. Palomar. Adaptive estimation of graphical models under total positivity. In Proceedings of the 40th International Conference on Machine Learning ( ICML 2023) , volume 202 of Proceedings of Machine Learning Research. PMLR, 2023

work page 2023