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Learning the Sherrington-Kirkpatrick Model Even at Low Temperature

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

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Pith's one-line read The paper claims SK spin-glass parameters are recoverable in polynomial time for every beta up to sqrt(log n), crossing the beta=1 phase transition.

desk verdict Breaking the β=1 barrier for learning the SK model is a real advance, but the paper as written has a repairable gap: the tail bound is proven for |A_i·X+h_iX_i| while the reduction needs |A_i·X+h_i|, and the external-field step is missing a bridge that a triangle inequality can fix. read the letter →

arxiv 2411.11174 v1 pith:GYEIGM3E submitted 2024-11-17 cs.LG cs.DSmath.STstat.MLstat.TH

Gautam Chandrasekaran , Adam Klivans This is my paper · ORCID
classification cs.LGcs.DSmath.STstat.MLstat.TH MSC 68Q3282B44
keywords Sherrington-KirkpatrickmodelIsinglearningparameterrecoverySparsitronmultiplicativeweightssubgaussianconcentrationlow-temperaturephaserandomMarkovfields
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 sets out to show that the Sherrington-Kirkpatrick (SK) model—an Ising model with iid Gaussian couplings scaled by $\beta/\sqrt{n}$—can be learned from samples even in the low-temperature phase $\beta\ge 1$, where earlier algorithms and the statistical-physics tools behind them stop working. It claims that running the Sparsitron multiplicative-weights algorithm coordinate by coordinate recovers the interaction matrix to entrywise accuracy $\epsilon$ and the external field to accuracy $n\epsilon$ using $N=\exp(O(\beta^2+\beta\sqrt{\log n}))\,\mathrm{poly}(n,1/\epsilon)$ samples and $O(n^2N)$ time; for $\beta\le\sqrt{\log n}$ this is polynomial time. If the claim is correct, the phase transition at $\beta=1$ is not a computational barrier for parameter recovery: the same simple algorithm works on both sides. The argument extends to random Ising models on arbitrary graphs and to higher-order random Markov random fields, where even high-temperature guarantees were previously open.

What carries the argument

Two ingredients carry the argument. The first is Sparsitron, a multiplicative-weights algorithm that learns a sigmoid function from labeled samples; the earlier reduction of parameter recovery to a squared-loss guarantee for this sigmoid is reused without change. The second is a new concentration lemma: the local field $A_i\cdot X+h_iX_i$, with $X$ drawn from the Gibbs measure, is subgaussian at scale $O(\beta)$ uniformly in $i$, with high probability over the random SK instance. The proof decouples the partition function by pairing each configuration $x$ with its spin-flip $x^{\{i\}}$ and uses the subgaussianity of the random couplings, so the exponential moment is controlled without any covariance operator norm. The resulting tail bound provides the constant-probability 'good region' in which the sigmoid is anti-Lipschitz with only a $\exp(O(\beta^2+\beta\sqrt{\log n}))$ penalty. For higher-order MRFs the same lemma is generalized to subgaussian differences $\psi(x)-\psi(y)$ of the factorization polynomial, which yields the $C$-smoothness condition used by the $t$-wise recovery theorems.

What would settle it

Run the claimed tail bound on finite SK instances: draw $D_{A,h}\sim SK(\beta)$ with $\beta=0.1$ and $n=1000$, sample $X$ from the model, and estimate $\Pr[|A_i\cdot X+h_iX_i| > C(\beta^2+\beta\sqrt{\log n})]$ for the constant $C$ in the $O(\cdot)$. If this probability is not below $1/n$ for every row, Lemma 3.4 fails as stated and the small-$\beta$ case would need a corrected proof. A complementary experiment at $\beta=1.5$ with the claimed $\exp(O(\beta^2))\,\mathrm{poly}(n)$ samples would test the main theorem directly.

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Extended reading notes

Core claim

The central discovery is that the low-temperature barrier for learning the SK model comes from the specific techniques used by prior work, not from the model itself. Previous approaches reduced learnability to a bound on the operator norm of the covariance $\mathbb{E}_{X\sim D_{A,h}}[XX^T]$, a quantity that provably diverges for $\beta\ge 1$. The paper instead proves a direct subgaussian bound on the local fields the algorithm actually needs: with high probability over the random choice of $A,h$, for every row $i$, $\mathbb{E}_{X\sim D_{A,h}}[\exp(|A_i\cdot X+h_iX_i|^2/(B\beta^2))]\le n^2 e^{O(\beta^2)}$. This yields a tail bound of the form $|A_i\cdot X+h_iX_i|\le O(\beta^2+\beta\sqrt{\log n})$ except with probability $1/n$, and this tail bound is exactly what the parameter-recovery reduction requires. Plugging it into the Sparsitron analysis produces the sample bound $N=\exp(O(\beta^2+\beta\sqrt{\log n}))\,\mathrm{poly}(n,1/\epsilon)$ and thus polynomial time for $\beta\le\sqrt{\log n}$.

Load-bearing premise

The load-bearing premise is that each local field is subgaussian at scale proportional to $\beta$, whereas a standard Gaussian external field has scale $1$; for small $\beta$ the proof's normalization in the exponential-moment lemma is therefore not justified as written, and the displayed calculation even switches between $A_i\cdot X+h_i$ and $A_i\cdot X+h_iX_i$.

Editorial extensions

If this is right

  • For any fixed inverse temperature $\beta>1$, the sample count becomes $\exp(O(\beta^2))\,\mathrm{poly}(n,1/\epsilon)$, so every fixed low temperature is learned in polynomial time, not just a thin band above $\beta=1$.
  • Success over the random model holds with probability $1-O(1/n)$, improving on the $1-O(1/\log n)$ guarantee of the prior high-temperature algorithm, and a Gaussian external field requires no extra treatment.
  • On arbitrary bounded-degree graphs with random Gaussian weights, the same algorithm recovers the graph and parameters with $\exp(O(\beta^2+\beta\sqrt{\log n}))\,\mathrm{poly}(d,\log n,1/\epsilon)$ samples; with Rademacher weights it recovers the model exactly with a sub-polynomial sample count.
  • For random $t$-wise MRFs and pure $t$-spin models, parameter recovery and TV-distance closeness hold with $\exp(O(\beta^2 t+\beta t\sqrt{\log n}))\,\mathrm{poly}(n^t,1/\epsilon)$ samples, a regime where even high-temperature results were previously open.
  • Because parameter recovery implies KL and TV closeness, the algorithm can output an Ising model $\epsilon$-close in total variation to the true one.

Reading between the lines

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

  • Editorial inference: the real condition suggested by the proof is subgaussian concentration of the Gibbs local fields, not boundedness of the covariance operator; other disordered models with such concentration, such as diluted spin glasses or mixtures of $p$-spin terms, should be learnable by the same argument even where spectral conditions fail.
  • Editorial inference: the bound $\beta\le\sqrt{\log n}$ is a sufficient condition, not a proven frontier; the exponential-moment calculation is loose enough that the true polynomial-time threshold may extend to larger $\beta$, and a finite-$n$ experiment at fixed $\beta=2$ is a natural probe.
  • Editorial inference: a fully uniform statement over all $\beta>0$ still needs one additional calculation, because the local-field moment lemma is normalized by $\beta$ while a standard Gaussian external field has scale $1$; for very small $\beta$ the lemma as written needs a separate argument.
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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

2 major / 4 minor

Summary. The paper studies parameter recovery for Ising models and higher-order Markov random fields with random coefficients. It proves that Sparsitron, run with N = exp(O(β² + β√log n)) · poly(n, 1/ε) samples, recovers the interaction matrix and external field of the Sherrington-Kirkpatrick model to accuracy ε in infinity norm (with a factor n for the external field) with probability 1 − O(1/n) over the random parameter draw. This covers β ≤ √log n, including the low-temperature regime β ≥ 1 where covariance-based arguments provably fail. The proof bounds tails of local fields via a partition-function decoupling argument and subgaussian concentration, then applies the Gaitonde-Mossel reduction. The paper also extends the approach to random MRFs on arbitrary bounded-degree graphs and to pure p-spin models, with exact recovery in the Rademacher case.

Significance. If the central argument is repaired, this is a substantial advance: it gives the first polynomial-time parameter recovery for the SK model past β = 1, with a proof that avoids the covariance-operator bounds and functional inequalities used in prior work. The extension to p-spin models and random MRFs on arbitrary graphs is also novel, as is the sub-polynomial exact recovery for Rademacher weights. The proof strategy is simple and appears to be correct in its main mechanism. I found no circularity, and the use of the KM17 and GM24 results as black boxes is appropriate. The technical gaps identified below are local and repairable, and for the headline low-temperature regime β ≥ 1 the indicated repair does not change the stated exponents.

major comments (2)
  1. [Section 3.3, Lemmas 3.3–3.4 and Theorem 3.5] Lemma 3.4 proves a tail bound on |A_i·X + h_i X_i|, but Theorem 3.2, which is the reduction invoked in the proof of Theorem 3.5, requires condition (2) to hold for |A_i·X + h_i|. These are not the same event: for a fixed nonzero h_i, a configuration with |A_i·X + h_i| > C can satisfy |A_i·X + h_i X_i| ≤ C when X_i has the sign that cancels h_i. The proof of Theorem 3.5 does not supply the missing step. A repair is available via |A_i·X + h_i| ≤ |A_i·X + h_i X_i| + 2|h_i| and the Gaussian tail |h_i| ≤ O(√log n), which yields C' = O(β² + β√log n) for β ≥ 1 but C' = O(√log n) for small β. The external-field recovery in Theorem 3.7 and the uniform-in-β sample complexity in Theorems 3.5 and 1.1 are therefore not established as written.
  2. [Section 3.3, Lemma 3.3] The proof claims that A_i·x + h_i is O(β)-subgaussian because h_i is O(1)-subgaussian. Since h_i ∼ N(0,1), the sum has variance β² + 1 and subgaussian constant Θ(max(β,1)), not Θ(β). Consequently the normalization Bβ² in Eq. (4) is invalid for β below a constant: already the h_i term alone gives E exp(h_i²/(Bβ²)) = ∞ for sufficiently small β, contradicting the claimed finite expectation. Replacing the denominator by B(β² + 1) repairs Lemma 3.3 but propagates to Lemma 3.4, changing the tail constant from O(β² + β√log n) to O(β² + 1 + √log n) for small β.
minor comments (4)
  1. [Theorem 1.1 vs. Theorem 3.7] The informal Theorem 1.1 states ‖h − ĥ‖∞ ≤ ε, whereas the formal Theorem 3.7 proves only ‖h − ĥ‖∞ ≤ nε. Please align the informal and formal statements, or track the extra n factor explicitly.
  2. [Definition B.2] The sentence introducing total variation distance says 'The KL-divergence between P and Q is defined as' but the displayed formula is the total variation distance; this should be corrected.
  3. [Theorem 3.7] The displayed sample complexity is O(β² n log n log(n/δε)/ε⁴), but substituting λ = O(β√n log n) into Theorem 3.1 gives an additional log n factor; please reconcile the exponents.
  4. [Remark 3.10] The parameter γ = √(βt + σ²) appears to contain a typo; the variance of the local field suggests γ should be of order √(β² + σ²) or the intended rescaling should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim is derived from independent concentration arguments and external black-box results, with only a minor non-load-bearing self-citation.

full rationale

I walked the derivation chain Lemma 3.3 -> Lemma 3.4 -> Theorem 3.5 -> Theorem 3.7. The load-bearing ingredients are Sparsitron from [KM17] (Theorem 3.1) and the reduction theorem of [GM24] (Theorem 3.2), both used as black boxes external to this paper. Although Adam Klivans is a coauthor of [KM17], that prior result is parameter-free, does not contain the low-temperature SK recovery conclusion, and is independently checkable, so under the stated rules it does not raise the circularity score. The only places where the paper relies on its own argument are the new moment and tail bounds for |Ai.X + hi Xi|, and these are not defined in terms of the output A_hat, h_hat or in terms of the sample complexity being claimed. The reader's noted mismatch—that Lemma 3.4 proves a tail bound for |Ai.X + hi Xi| while Theorem 3.2 from [GM24] requires a tail bound for |Ai.X + hi|—and the small-beta issue with the claimed O(beta)-subgaussian constant are genuine correctness gaps in the logical chain, but they are not circularity: failing to prove a hypothesis is not the same as assuming the conclusion or renaming a fitted input as a prediction. No equation in the paper makes the recovered parameters equal, by construction, to the quantities that were fit or assumed, and no load-bearing step reduces to a self-citation chain.

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

The core of the paper is the new Lemma 3.3; all other ingredients are prior theorems or standard analysis. The most questionable implicit assumption is the O(beta) subgaussian scale for |Ai.X+hi| when hi has variance 1, which is not separately stated and is partially contradicted by Remark 3.10.

assumptions (5)
  • standard math Sparsitron (KM17, Theorem 3.1) learns sigmoid GLMs with sample complexity O(lambda^2 log(n/delta/epsilon)/epsilon^2).
    Used as the base learner; the paper's runtime and sample bounds inherit this.
  • standard math Gaitonde-Mossel (GM24, Theorem 3.2) converts squared sigmoid loss to l-infinity parameter error under conditions (1) ||Ai||_inf <= C and (2) Pr[|Ai.X+hi| <= C] >= 3/4.
    Reduction that turns the new concentration bound into parameter recovery; cited as prior work.
  • standard math Subgaussian MGF bounds and the anti-Lipschitz property of the sigmoid (Facts 2.1 and 2.2).
    Elementary analytic tools used throughout.
  • domain assumption The random Ising/MRF weight distributions (Gaussian or Rademacher) as defined in Definitions 1.3 and 1.11.
    The learning guarantees are stated relative to these distributions.
  • standard math The partition function lower bound Z >= (1/2) sum_x exp(g_{-i}(x)) via e^t + e^{-t} >= 1.
    Used in Lemma 3.3 to decouple row i from the normalization.

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Cite this review

Pith. "Pith review of Learning the Sherrington-Kirkpatrick Model Even at Low Temperature." pith.science (2026). https://pith.science/paper/GYEIGM3E

@misc{pith2026241111174,
  author       = {Pith},
  title        = {Pith review of: Learning the Sherrington-Kirkpatrick Model Even at Low Temperature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GYEIGM3E}},
  note         = {Machine review of arXiv:2411.11174}
}
abstract

We consider the fundamental problem of learning the parameters of an undirected graphical model or Markov Random Field (MRF) in the setting where the edge weights are chosen at random. For Ising models, we show that a multiplicative-weight update algorithm due to Klivans and Meka learns the parameters in polynomial time for any inverse temperature $\beta \leq \sqrt{\log n}$. This immediately yields an algorithm for learning the Sherrington-Kirkpatrick (SK) model beyond the high-temperature regime of $\beta < 1$. Prior work breaks down at $\beta = 1$ and requires heavy machinery from statistical physics or functional inequalities. In contrast, our analysis is relatively simple and uses only subgaussian concentration. Our results extend to MRFs of higher order (such as pure $p$-spin models), where even results in the high-temperature regime were not known.

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Reference graph

Works this paper leans on

43 extracted references · 15 canonical work pages · cited by 1 Pith paper

  1. [1]

    Spectral gap estimates for mixed p-spin models at high temperature

    Arka Adhikari, Christian Brennecke, Changji Xu, and Horng-Tzer Yau. Spectral gap estimates for mixed p-spin models at high temperature. Probability Theory and Related Fields , pages 1--29, 2024

    work page 2024

  2. [2]

    Shattering versus metastability in spin glasses

    G \'e rard Ben Arous and Aukosh Jagannath. Shattering versus metastability in spin glasses. Communications on Pure and Applied Mathematics , 77(1):139--176, 2024

    work page 2024

  3. [3]

    Universality of spectral independence with applications to fast mixing in spin glasses

    Nima Anari, Vishesh Jain, Frederic Koehler, Huy Tuan Pham, and Thuy-Duong Vuong. Universality of spectral independence with applications to fast mixing in spin glasses. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) , pages 5029--5056. SIAM, 2024

    work page 2024

  4. [4]

    Learning factor graphs in polynomial time and sample complexity

    Pieter Abbeel, Daphne Koller, and Andrew Y Ng. Learning factor graphs in polynomial time and sample complexity. The Journal of Machine Learning Research , 7:1743--1788, 2006

    2006

  5. [5]

    Shattering in pure spherical spin glasses

    Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Shattering in pure spherical spin glasses. arXiv preprint arXiv:2307.04659 , 2023

    arXiv 2023

  6. [6]

    A very simple proof of the lsi for high temperature spin systems

    Roland Bauerschmidt and Thierry Bodineau. A very simple proof of the lsi for high temperature spin systems. Journal of Functional Analysis , 276(8):2582--2588, 2019

    work page 2019

  7. [7]

    Near-optimal learning of tree-structured distributions by chow-liu

    Arnab Bhattacharyya, Sutanu Gayen, Eric Price, and NV Vinodchandran. Near-optimal learning of tree-structured distributions by chow-liu. In Proceedings of the 53rd annual acm SIGACT symposium on theory of computing , pages 147--160, 2021

    2021

  8. [8]

    Reconstruction of markov random fields from samples: Some observations and algorithms

    Guy Bresler, Elchanan Mossel, and Allan Sly. Reconstruction of markov random fields from samples: Some observations and algorithms. In International Workshop on Approximation Algorithms for Combinatorial Optimization , pages 343--356. Springer, 2008

    2008

Show all 43 references
  1. [9]

    Efficiently learning ising models on arbitrary graphs

    Guy Bresler. Efficiently learning ising models on arbitrary graphs. In Rocco A. Servedio and Ronitt Rubinfeld, editors, Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015 , pages 771--782. ACM , 2015

    2015

  2. [10]

    A statistical physics approach to learning curves for the inverse ising problem

    Ludovica Bachschmid-Romano and Manfred Opper. A statistical physics approach to learning curves for the inverse ising problem. Journal of Statistical Mechanics: Theory and Experiment , 2017(6):063406, 2017

    2017

  3. [11]

    The two point function of the sk model without external field at high temperature

    Christian Brennecke, Adrien Schertzer, Changji Xu, and Horng-Tzer Yau. The two point function of the sk model without external field at high temperature. Probab. Math. Phys , 5:131--175, 2024

    2024

  4. [12]

    Operator norm bounds on the correlation matrix of the sk model at high temperature

    Christian Brennecke, Changji Xu, and Horng-Tzer Yau. Operator norm bounds on the correlation matrix of the sk model at high temperature. arXiv preprint arXiv:2307.12535 , 2023

    2023 arXiv

  5. [13]

    Markov fields on finite graphs and lattices

    P Clifford and JM Hammersley. Markov fields on finite graphs and lattices. 1971

    1971

  6. [14]

    Approximating discrete probability distributions with dependence trees

    CKCN Chow and Cong Liu. Approximating discrete probability distributions with dependence trees. IEEE transactions on Information Theory , 14(3):462--467, 1968

    1968

  7. [15]

    Learning ising models from one or multiple samples

    Yuval Dagan, Constantinos Daskalakis, Nishanth Dikkala, and Anthimos Vardis Kandiros. Learning ising models from one or multiple samples. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing , pages 161--168, 2021

    2021

  8. [16]

    Outlier-robust learning of ising models under dobrushin’s condition

    Ilias Diakonikolas, Daniel M Kane, Alistair Stewart, and Yuxin Sun. Outlier-robust learning of ising models under dobrushin’s condition. In Conference on Learning Theory , pages 1645--1682. PMLR, 2021

    2021

  9. [17]

    Theory of spin glasses

    Samuel Frederick Edwards and Phil W Anderson. Theory of spin glasses. Journal of Physics F: Metal Physics , 5(5):965, 1975

    1975

  10. [18]

    Bounds on the covariance matrix of the sherrington--kirkpatrick model

    Ahmed El Alaoui and Jason Gaitonde. Bounds on the covariance matrix of the sherrington--kirkpatrick model. Electronic Communications in Probability , 29:1--13, 2024

    2024

  11. [19]

    A spectral condition for spectral gap: fast mixing in high-temperature ising models

    Ronen Eldan, Frederic Koehler, and Ofer Zeitouni. A spectral condition for spectral gap: fast mixing in high-temperature ising models. Probability theory and related fields , 182(3):1035--1051, 2022

    2022

  12. [20]

    Shattering in the ising pure p -spin model

    David Gamarnik, Aukosh Jagannath, and Eren C K z lda g . Shattering in the ising pure p -spin model. arXiv preprint arXiv:2307.07461 , 2023

    2023 arXiv

  13. [21]

    Learning ising models with independent failures

    Surbhi Goel, Daniel M Kane, and Adam R Klivans. Learning ising models with independent failures. In Conference on Learning Theory , pages 1449--1469. PMLR, 2019

    2019

  14. [22]

    A unified approach to learning ising models: Beyond independence and bounded width

    Jason Gaitonde and Elchanan Mossel. A unified approach to learning ising models: Beyond independence and bounded width. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing , pages 503--514, 2024

    2024

  15. [23]

    Efficiently learning markov random fields from dynamics

    Jason Gaitonde, Ankur Moitra, and Elchanan Mossel. Efficiently learning markov random fields from dynamics. arXiv preprint arXiv:2409.05284 , 2024

    2024 arXiv

  16. [24]

    Information theoretic properties of markov random fields, and their algorithmic applications

    Linus Hamilton, Frederic Koehler, and Ankur Moitra. Information theoretic properties of markov random fields, and their algorithmic applications. Advances in Neural Information Processing Systems , 30, 2017

    2017

  17. [25]

    Efficient learning of generalized linear and single index models with isotonic regression

    Sham M Kakade, Varun Kanade, Ohad Shamir, and Adam Kalai. Efficient learning of generalized linear and single index models with isotonic regression. Advances in Neural Information Processing Systems , 24, 2011

    2011

  18. [26]

    Klivans and Raghu Meka

    Adam R. Klivans and Raghu Meka. Learning graphical models using multiplicative weights. 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) , pages 343--354, 2017

    2017

  19. [27]

    Information, physics, and computation

    Marc Mezard and Andrea Montanari. Information, physics, and computation . Oxford University Press, 2009

    2009

  20. [28]

    Constraint satisfaction problems and neural networks: A statistical physics perspective

    Marc M \'e zard and Thierry Mora. Constraint satisfaction problems and neural networks: A statistical physics perspective. Journal of Physiology-Paris , 103(1-2):107--113, 2009

    2009

  21. [29]

    Learning to sample from censored markov random fields

    Ankur Moitra, Elchanan Mossel, and Colin P Sandon. Learning to sample from censored markov random fields. In Conference on Learning Theory , pages 3419--3451. PMLR, 2021

    2021

  22. [30]

    Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications , volume 9

    Marc M \'e zard, Giorgio Parisi, and Miguel Angel Virasoro. Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications , volume 9. World Scientific Publishing Company, 1987

    1987

  23. [31]

    Greedy learning of markov network structure

    Praneeth Netrapalli, Siddhartha Banerjee, Sujay Sanghavi, and Sanjay Shakkottai. Greedy learning of markov network structure. In 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton) , pages 1295--1302. IEEE, 2010

    2010

  24. [32]

    The sherrington-kirkpatrick model

    Dmitry Panchenko. The sherrington-kirkpatrick model . Springer Science & Business Media, 2013

    2013

  25. [33]

    On learning ising models under huber's contamination model

    Adarsh Prasad, Vishwak Srinivasan, Sivaraman Balakrishnan, and Pradeep Ravikumar. On learning ising models under huber's contamination model. Advances in neural information processing systems , 33:16327--16338, 2020

    2020

  26. [34]

    Information-theoretic limits of selecting binary graphical models in high dimensions

    Narayana P Santhanam and Martin J Wainwright. Information-theoretic limits of selecting binary graphical models in high dimensions. IEEE Transactions on Information Theory , 58(7):4117--4134, 2012

    2012

  27. [35]

    Spin glasses: a challenge for mathematicians: cavity and mean field models , volume 46

    Michel Talagrand. Spin glasses: a challenge for mathematicians: cavity and mean field models , volume 46. Springer Science & Business Media, 2003

    2003

  28. [36]

    Mean field models for spin glasses: Volume I: Basic examples , volume 54

    Michel Talagrand. Mean field models for spin glasses: Volume I: Basic examples , volume 54. Springer Science & Business Media, 2010

    2010

  29. [37]

    Learning graphs with a few hubs

    Rashish Tandon and Pradeep Ravikumar. Learning graphs with a few hubs. In International conference on machine learning , pages 602--610. PMLR, 2014

    2014

  30. [38]

    Vershynin

    R. Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science . Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2018

    2018

  31. [39]

    Efficient learning of discrete graphical models

    Marc Vuffray, Sidhant Misra, and Andrey Y Lokhov. Efficient learning of discrete graphical models. Journal of Statistical Mechanics: Theory and Experiment , 2021(12):124017, 2022

    2021

  32. [40]

    Lokhov, and Michael Chertkov

    Marc Vuffray, Sidhant Misra, Andrey Y. Lokhov, and Michael Chertkov. Interaction screening: Efficient and sample-optimal learning of ising models. In Daniel D. Lee, Masashi Sugiyama, Ulrike von Luxburg, Isabelle Guyon, and Roman Garnett, editors, Advances in Neural Information...

    2016

  33. [41]

    High-dimensional graphical model selection using _1 -regularized logistic regression

    Martin J Wainwright, John Lafferty, and Pradeep Ravikumar. High-dimensional graphical model selection using _1 -regularized logistic regression. Advances in neural information processing systems , 19, 2006

    2006

  34. [42]

    Sparse logistic regression learns all discrete pairwise graphical models

    Shanshan Wu, Sujay Sanghavi, and Alexandros G Dimakis. Sparse logistic regression learns all discrete pairwise graphical models. Advances in Neural Information Processing Systems , 32, 2019

    2019

  35. [43]

    Privately learning markov random fields

    Huanyu Zhang, Gautam Kamath, Janardhan Kulkarni, and Steven Wu. Privately learning markov random fields. In International conference on machine learning , pages 11129--11140. PMLR, 2020

    2020

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