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arxiv: 2408.02360 · v3 · pith:VEBFGSIKnew · submitted 2024-08-05 · 🧮 math.PR · cs.DS· math-ph· math.MP

Potential Hessian Ascent: The Sherrington-Kirkpatrick Model

David Jekel , Juspreet Singh Sandhu , Jonathan Shi This is my paper

Pith reviewed 2026-05-23 22:37 UTC · model grok-4.3

classification 🧮 math.PR cs.DSmath-phmath.MP
keywords Sherrington-Kirkpatrick modelHessian ascentAuffinger-Chen SDEParisi PDEfree probabilityspin glasseshypercube optimizationquadratic objective
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The pith

A randomized Hessian ascent algorithm with potential modification finds near-optimal solutions to the Sherrington-Kirkpatrick model on the discrete hypercube.

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

The paper develops the first iterative spectral algorithm for near-optimal solutions to a random quadratic objective over the hypercube, resolving a conjecture of Subag. The procedure performs randomized ascent on an objective modified by an instance-independent potential function. It constructs an approximate projector onto the top Hessian eigenspaces via free probability theory to serve as the covariance for random increments. With high probability the iterates track the primal Auffinger-Chen SDE, and the per-step improvement is bounded by Taylor expansion under Gaussian concentration and smoothness of a semiconcave regularization tied to the Parisi PDE.

Core claim

The algorithm is a randomized Hessian ascent in the solid cube with the objective modified by subtracting an instance-independent potential. Using tools from free probability theory an approximate projector into the top eigenspaces of the Hessian is constructed and used as the covariance matrix for the random increments. With high probability the iterates' empirical distribution approximates the solution to the primal version of the Auffinger-Chen SDE. The per-iterate change in the modified objective is bounded via a Taylor expansion controlled by Gaussian concentration bounds and smoothness properties of a semiconcave regularization of the Fenchel-Legendre dual to the Parisi PDE.

What carries the argument

randomized Hessian ascent using an approximate projector into the top eigenspaces of the Hessian (via free probability) as covariance for random increments

If this is right

  • The algorithm reaches near-optimal values with high probability.
  • The empirical distribution of iterates approximates the primal Auffinger-Chen SDE.
  • The per-iterate change in the modified objective is bounded via Taylor expansion, Gaussian concentration, and smoothness of the semiconcave regularization.
  • The results provide groundwork for low-degree sum-of-squares certificates over high-entropy step distributions for a relaxed version of the Parisi formula.

Where Pith is reading between the lines

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

  • The same spectral-projection idea could be tested on other mean-field spin-glass Hamiltonians beyond the Sherrington-Kirkpatrick case.
  • If the SDE approximation is stable under discretization, the method might yield practical heuristics for moderate-sized instances of related quadratic optimization problems.
  • The potential-modification technique may connect to other regularized ascent schemes that avoid explicit solution of the Parisi PDE.
  • Whether the same free-probability projector construction extends to non-Gaussian disorder remains open.

Load-bearing premise

The iterates' empirical distribution approximates the solution to the primal Auffinger-Chen SDE closely enough for the Taylor expansion bounds to hold.

What would settle it

Numerical simulation on large instances showing that the empirical distribution of iterates deviates substantially from the predicted distribution of the primal Auffinger-Chen SDE would falsify the tracking claim.

read the original abstract

We present the first iterative spectral algorithm to find near-optimal solutions for a random quadratic objective over the discrete hypercube, resolving a conjecture of Subag [Subag, Communications on Pure and Applied Mathematics, 74(5), 2021]. The algorithm is a randomized Hessian ascent in the solid cube, with the objective modified by subtracting an instance-independent potential function [Chen et al., Communications on Pure and Applied Mathematics, 76(7), 2023]. Using tools from free probability theory, we construct an approximate projector into the top eigenspaces of the Hessian, which serves as the covariance matrix for the random increments. With high probability, the iterates' empirical distribution approximates the solution to the primal version of the Auffinger-Chen SDE [Auffinger et al., Communications in Mathematical Physics, 335, 2015]. The per-iterate change in the modified objective is bounded via a Taylor expansion, where the derivatives are controlled through Gaussian concentration bounds and smoothness properties of a semiconcave regularization of the Fenchel-Legendre dual to the Parisi PDE. These results lay the groundwork for (possibly) demonstrating low-degree sum-of-squares certificates over high-entropy step distributions for a relaxed version of the Parisi formula [Open Question 1.8, arXiv:2401.14383].

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.

Referee Report

0 major / 2 minor

Summary. The manuscript presents the first iterative spectral algorithm for finding near-optimal solutions to a random quadratic objective (the Sherrington-Kirkpatrick model) over the discrete hypercube, thereby resolving a conjecture of Subag. The algorithm performs randomized Hessian ascent inside the solid cube on an objective modified by an instance-independent potential function. Free probability is used to construct an approximate projector onto the top eigenspaces of the Hessian, which serves as the covariance for the random increments. With high probability the empirical distribution of the iterates tracks the solution of the primal Auffinger-Chen SDE; the per-iterate change in the modified objective is controlled by a Taylor expansion whose error is bounded via Gaussian concentration together with the semiconcavity and smoothness of a regularized Fenchel-Legendre dual to the Parisi PDE. The work also indicates a possible route toward low-degree sum-of-squares certificates for a relaxed Parisi formula.

Significance. If the detailed error controls and approximation statements hold, the result would constitute a substantial advance: it supplies the first iterative method that provably reaches near-optimal values for the SK model on the hypercube and resolves Subag's conjecture. The synthesis of free-probability constructions, SDE tracking, and PDE-derived regularity estimates is technically novel and directly addresses an open algorithmic question in spin-glass theory. The groundwork laid for sum-of-squares certificates is a further positive feature.

minor comments (2)
  1. [Abstract] Abstract, final paragraph: the parenthetical '(possibly)' in the sentence on sum-of-squares certificates should be replaced by a precise statement of what is proved versus what remains open.
  2. [Introduction] The notation for the modified objective function and the precise definition of the semiconcave regularization should be introduced with a displayed equation early in the introduction to improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are pleased that the referee finds the result a substantial advance and recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation chain relies on external, independently established results (free probability for the projector, Auffinger-Chen SDE for the empirical measure, Parisi PDE dual for the Taylor error bounds) cited from prior literature with no author overlap. No step reduces a claimed prediction or uniqueness statement to a quantity defined or fitted inside the paper itself; the central algorithm and its analysis remain self-contained against those external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the paper relies on domain assumptions from free probability and PDE theory whose precise statements and applicability are not inspectable here.

axioms (2)
  • domain assumption Free probability theory yields an approximate projector onto the top eigenspaces of the Hessian that can serve as covariance for random increments.
    Invoked to define the random steps of the ascent.
  • domain assumption The semiconcave regularization of the Fenchel-Legendre dual to the Parisi PDE possesses sufficient smoothness for derivative control in the Taylor expansion.
    Required to bound the per-iterate objective change via Gaussian concentration.

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Works this paper leans on

86 extracted references · 86 canonical work pages · 3 internal anchors

  1. [1]

    write newline

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    work page

  2. [2]

    Lectures on optimal transport , volume 130

    Luigi Ambrosio, Elia Bru \'e , and Daniele Semola. Lectures on optimal transport , volume 130. Springer, 2021

    work page 2021

  3. [3]

    On properties of Parisi measures

    Antonio Auffinger and Wei-Kuo Chen. On properties of Parisi measures. Probability Theory and Related Fields , 161(3):817--850, 2015

    work page 2015

  4. [4]

    The Parisi formula has a unique minimizer

    Antonio Auffinger and Wei-Kuo Chen. The Parisi formula has a unique minimizer. Communications in Mathematical Physics , 335(3):1429--1444, 2015

    work page 2015

  5. [5]

    Parisi formula for the ground state energy in the mixed p -spin model

    Antonio Auffinger and Wei-Kuo Chen. Parisi formula for the ground state energy in the mixed p -spin model . The Annals of Probability , 45(6B):4617 -- 4631, 2017

    work page 2017

  6. [6]

    The SK model is infinite step replica symmetry breaking at zero temperature

    Antonio Auffinger, Wei-Kuo Chen, and Qiang Zeng. The SK model is infinite step replica symmetry breaking at zero temperature. Communications on Pure and Applied Mathematics , 73(5), 2020

    work page 2020

  7. [7]

    An Introduction to II_1 Factors

    Claire Anantharaman-Delaroche and Sorin Popa. An Introduction to II_1 Factors. Preprint available at https://www.math.ucla.edu/\ popa/Books/IIunV15.pdf , 2021

    work page 2021

  8. [8]

    N. I. Akhiezer and I. M. Glazman. Theory of Linear Operators in Hilbert Space . Ungar, 1963

    work page 1963

  9. [9]

    An introduction to random matrices

    Greg W Anderson, Alice Guionnet, and Ofer Zeitouni. An introduction to random matrices . Cambridge university press, 2010

    work page 2010

  10. [10]

    Optimization of mean-field spin glasses

    Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Optimization of mean-field spin glasses. arXiv preprint arXiv:2001.00904 , 2020

    work page arXiv 2001

  11. [11]

    Local algorithms for Maximum Cut and Minimum Bisection on locally treelike regular graphs of large degree

    Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Local algorithms for Maximum Cut and Minimum Bisection on locally treelike regular graphs of large degree. arXiv preprint arXiv:2111.06813 , 2021

    work page arXiv 2021

  12. [12]

    Extended variational principle for the Sherrington-Kirkpatrick spin-glass model

    Michael Aizenman, Robert Sims, and Shannon L Starr. Extended variational principle for the Sherrington-Kirkpatrick spin-glass model. Physical Review B , 68(21):214403, 2003

    work page 2003

  13. [13]

    Belinschi and Hari Bercovici

    Serban T. Belinschi and Hari Bercovici. A new approach to subordination results in free probability. J Anal Math , 101:357--365, 2007

    work page 2007

  14. [14]

    Belinschi

    Serban T. Belinschi. Complex analysis methods in non-commutative probability . PhD thesis, Indiana University, 2005

    work page 2005

  15. [15]

    Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere

    Vijay Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Sum-of-squares certificates for maxima of random tensors on the sphere. arXiv preprint arXiv:1605.00903 , 2016

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  16. [16]

    Processes with free increments

    Philippe Biane. Processes with free increments. Math. Z. , 227:143--174, 1998

    work page 1998

  17. [17]

    S. G. Bobkov and M. Ledoux. From B runn- M inkowski to B raskamp- L ieb and to logarithmic S obolev inequalities. Geom. Funct. Anal. , 10:1028--1052, 2000

    work page 2000

  18. [18]

    Operator Algebras: Theory of C ^* -algebras and von N eumann algebras , volume 122 of Encyclopaedia of Mathematical Sciences

    Bruce Blackadar. Operator Algebras: Theory of C ^* -algebras and von N eumann algebras , volume 122 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, Heidelberg, 2006

    work page 2006

  19. [19]

    The LASSO risk for Gaussian matrices

    Mohsen Bayati and Andrea Montanari. The LASSO risk for Gaussian matrices. IEEE Transactions on Information Theory , 58(4):1997--2017, 2011

    work page 1997

  20. [20]

    Proofs, beliefs, and algorithms through the lens of sum-of-squares

    Boaz Barak and David Steurer. Proofs, beliefs, and algorithms through the lens of sum-of-squares. Course notes: http://www. sumofsquares. org/public/index. html , 1, 2016

    work page 2016

  21. [21]

    Suboptimality of local algorithms for a class of max-cut problems

    Wei-Kuo Chen, David Gamarnik, Dmitry Panchenko, Mustazee Rahman, et al. Suboptimality of local algorithms for a class of max-cut problems. Annals of Probability , 47(3):1587--1618, 2019

    work page 2019

  22. [22]

    On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices

    Beno \^i t Collins, Alice Guionnet, and F \'e lix Parraud. On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices. Cambridge Journal of Mathematics , 10(1):195--260, 2022

    work page 2022

  23. [23]

    Variational representations for the parisi functional and the two-dimensional Guerra-Talagrand bound

    Wei Kuo Chen. Variational representations for the parisi functional and the two-dimensional Guerra-Talagrand bound. Annals of Probability , 45(6):3929--3966, 2017

    work page 2017

  24. [24]

    Local algorithms and the failure of log-depth quantum advantage on sparse random CSPs

    Antares Chen, Neng Huang, and Kunal Marwaha. Local algorithms and the failure of log-depth quantum advantage on sparse random CSPs. arXiv preprint arXiv:2310.01563 , 2023

    work page arXiv 2023

  25. [25]

    Limitations of local quantum algorithms on random max-k-xor and beyond

    Chi-Ning Chou, Peter J Love, Juspreet Singh Sandhu, and Jonathan Shi. Limitations of local quantum algorithms on random max-k-xor and beyond. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) . Schloss Dagstuhl-Leibniz-Zentrum f \"u r Informatik, 2022

    work page 2022

  26. [26]

    Fundamental barriers to high-dimensional regression with convex penalties

    Michael Celentano and Andrea Montanari. Fundamental barriers to high-dimensional regression with convex penalties. The Annals of Statistics , 50(1):170--196, 2022

    work page 2022

  27. [27]

    Generalized TAP Free Energy

    Wei-Kuo Chen, Dmitry Panchenko, and Eliran Subag. Generalized TAP Free Energy. Communications on Pure and Applied Mathematics , 2018

    work page 2018

  28. [28]

    Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models

    Wei-Kuo Chen and Arnab Sen. Parisi formula, disorder chaos and fluctuation for the ground state energy in the spherical mixed p-spin models. Communications in Mathematical Physics , 350(1):129--173, 2017

    work page 2017

  29. [29]

    Lecture Notes on Noncommutative Lp-Spaces

    Ricardo Correa da Silva . Lecture Notes on Non-commutative L_p -spaces. arXiv:1803.02390, 2018

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  30. [30]

    Formes lin \'e aires sur un anneau d'op \'e rateurs

    Jacques Dixmier. Formes lin \'e aires sur un anneau d'op \'e rateurs. Bulletin de la Soci \'e t \'e Math \'e matique de France , 81:9--39, 1953

    work page 1953

  31. [31]

    Message-passing algorithms for compressed sensing

    David L Donoho, Arian Maleki, and Andrea Montanari. Message-passing algorithms for compressed sensing. Proceedings of the National Academy of Sciences , 106(45):18914--18919, 2009

    work page 2009

  32. [32]

    Multilinear function series and transforms in free probability theory

    Kenneth J Dykema. Multilinear function series and transforms in free probability theory. Advances in Mathematics , 208(1):351--407, 2007

    work page 2007

  33. [33]

    Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization

    Ahmed El Alaoui, Andrea Montanari, and Mark Sellke. Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS) , pages 323--334. IEEE, 2022

    work page 2022

  34. [34]

    The overlap gap property and approximate message passing algorithms for p -spin models

    David Gamarnik and Aukosh Jagannath. The overlap gap property and approximate message passing algorithms for p -spin models . The Annals of Probability , 49(1):180 -- 205, 2021

    work page 2021

  35. [35]

    Sum-of-squares lower bounds for sherrington-kirkpatrick via planted affine planes

    Mrinalkanti Ghosh, Fernando Granha Jeronimo, Chris Jones, Aaron Potechin, and Goutham Rajendran. Sum-of-squares lower bounds for sherrington-kirkpatrick via planted affine planes. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) , pages 954--965. IEEE, 2020

    work page 2020

  36. [36]

    Low-degree hardness of random optimization problems

    David Gamarnik, Aukosh Jagannath, and Alexander S Wein. Low-degree hardness of random optimization problems. arXiv preprint arXiv:2004.12063 , 2020

    work page arXiv 2004

  37. [37]

    Long time behaviour of the solution to non-linear Kraichnan equations

    Alice Guionnet and Christian Mazza. Long time behaviour of the solution to non-linear Kraichnan equations. Probab. Theory Relat. Fields , 131:493–518, 2005

    work page 2005

  38. [38]

    Limits of local algorithms over sparse random graphs

    David Gamarnik and Madhu Sudan. Limits of local algorithms over sparse random graphs. In Proceedings of the 5th conference on Innovations in theoretical computer science , pages 369--376, 2014

    work page 2014

  39. [39]

    On the concavity of the TAP free energy in the SK model

    Stephan Gufler, Adrien Schertzer, and Marius A Schmidt. On the concavity of the TAP free energy in the SK model. Stochastic Processes and their Applications , 164:160--182, 2023

    work page 2023

  40. [40]

    The thermodynamic limit in mean field spin glass models

    Francesco Guerra and Fabio Lucio Toninelli. The thermodynamic limit in mean field spin glass models. Communications in Mathematical Physics , 230(1):71--79, 2002

    work page 2002

  41. [41]

    Sum rules for the free energy in the mean field spin glass model

    Francesco Guerra. Sum rules for the free energy in the mean field spin glass model. Fields Institute Communications , 30(11), 2001

    work page 2001

  42. [42]

    Broken replica symmetry bounds in the mean field spin glass model

    Francesco Guerra. Broken replica symmetry bounds in the mean field spin glass model. Communications in mathematical physics , 233:1--12, 2003

    work page 2003

  43. [43]

    The power of sum-of-squares for detecting hidden structures

    Samuel B Hopkins, Pravesh K Kothari, Aaron Potechin, Prasad Raghavendra, Tselil Schramm, and David Steurer. The power of sum-of-squares for detecting hidden structures. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS) , pages 720--731. IEEE, 2017

    work page 2017

  44. [44]

    Mirrored langevin dynamics

    Ya-Ping Hsieh, Ali Kavis, Paul Rolland, and Volkan Cevher. Mirrored langevin dynamics. Advances in Neural Information Processing Systems , 31, 2018

    work page 2018

  45. [45]

    Sampling from spherical spin glasses in total variation via algorithmic stochastic localization

    Brice Huang, Andrea Montanari, and Huy Tuan Pham. Sampling from spherical spin glasses in total variation via algorithmic stochastic localization. arXiv preprint arXiv:2404.15651 , 2024

    work page arXiv 2024

  46. [46]

    Huang and M

    Brice Huang and Mark Sellke. Tight Lipschitz hardness for optimizing mean field spin glasses. arXiv preprint arXiv:2110.07847 , 2021

    work page arXiv 2021

  47. [47]

    Upgraded free independence phenomena for random unitaries

    David Jekel and Srivatsav Kunnawalkam Elayavalli . Upgraded free independence phenomena for random unitaries. Preprint, arXiv:2404.17114, 2024

    work page arXiv 2024

  48. [48]

    Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

    Chris Jones, Kunal Marwaha, Juspreet Singh Sandhu, and Jonathan Shi. Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses. arXiv preprint arXiv:2210.03006 , 2022

    work page arXiv 2022

  49. [49]

    Jones and V

    V. Jones and V. S. Sunder. Introduction to Subfactors . London Mathematical Society Lecture Note Series. Cambridge University Press, 1997

    work page 1997

  50. [50]

    A dynamic programming approach to the Parisi functional

    Aukosh Jagannath and Ian Tobasco. A dynamic programming approach to the Parisi functional. Proceedings of the American Mathematical Society , 144(7):3135--3150, 2016

    work page 2016

  51. [51]

    Kadison and John R

    Richard V. Kadison and John R. Ringrose. Fundamentals of the Theory of Operator Algebras I , volume 15 of Graduate Studies in Mathematics . American Mathematical Society, Providence, 1983

    work page 1983

  52. [52]

    A heat semigroup approach to concentration on the sphere and on a compact R iemannian manifold

    Michel Ledoux. A heat semigroup approach to concentration on the sphere and on a compact R iemannian manifold. Geometric & Functional Analysis , 2(2):221--224, 06 1992

    work page 1992

  53. [53]

    The concentration of measure phenomenon , volume 89 of Mathematical Surveys and Monographs

    Michel Ledoux. The concentration of measure phenomenon , volume 89 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2001

    work page 2001

  54. [54]

    A Remark on Hypercontractivity and Tail Inequalities for the Largest Eigenvalues of Random Matrices , pages 360--369

    Michel Ledoux. A Remark on Hypercontractivity and Tail Inequalities for the Largest Eigenvalues of Random Matrices , pages 360--369. Springer Berlin Heidelberg, Berlin, Heidelberg, 2003

    work page 2003

  55. [55]

    Montanari

    A. Montanari. Optimization of the Sherrington-Kirkpatrick Hamiltonian. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS) , pages 1417--1433, Los Alamitos, CA, USA, 2019. IEEE Computer Society

    work page 2019

  56. [56]

    Optimization of the SK Model, LIDS Student Seminar, MIT, 2019

    Andrea Montanari. Optimization of the SK Model, LIDS Student Seminar, MIT, 2019

    work page 2019

  57. [57]

    Un-inverting the Parisi formula

    Jean-Christophe Mourrat. Un-inverting the Parisi formula. arXiv preprint arXiv:2308.10715 , 2023

    work page arXiv 2023

  58. [58]

    Free probability and random matrices , volume 35

    James A Mingo and Roland Speicher. Free probability and random matrices , volume 35. Springer, 2017

    work page 2017

  59. [59]

    Estimation of low-rank matrices via approximate message passing

    Andrea Montanari and Ramji Venkataramanan. Estimation of low-rank matrices via approximate message passing . The Annals of Statistics , 49(1):321 -- 345, 2021

    work page 2021

  60. [60]

    Lectures on the combinatorics of free probability , volume 13

    Alexandru Nica and Roland Speicher. Lectures on the combinatorics of free probability , volume 13. Cambridge University Press, 2006

    work page 2006

  61. [61]

    Stochastic differential equations: an introduction with applications

    Bernt Oksendal. Stochastic differential equations: an introduction with applications . Springer Science & Business Media, 2013

    work page 2013

  62. [62]

    The Parisi ultrametricity conjecture

    Dmitry Panchenko. The Parisi ultrametricity conjecture. Annals of Mathematics , pages 383--393, 2013

    work page 2013

  63. [63]

    The sherrington-kirkpatrick model

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

    work page 2013

  64. [64]

    Introduction to the SK model

    Dmitry Panchenko. Introduction to the SK model. arXiv preprint arXiv:1412.0170 , 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  65. [65]

    A sequence of approximated solutions to the SK model for spin glasses

    Giorgio Parisi. A sequence of approximated solutions to the SK model for spin glasses. Journal of Physics A: Mathematical and General , 13(4):L115, 1980

    work page 1980

  66. [66]

    Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

    F \'e lix Parraud. Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices. Communications in Mathematical Physics , 399(1):249--294, 2023

    work page 2023

  67. [67]

    Fast matrix square roots with applications to Gaussian processes and Bayesian optimization

    Geoff Pleiss, Martin Jankowiak, David Eriksson, Anil Damle, and Jacob Gardner. Fast matrix square roots with applications to Gaussian processes and Bayesian optimization. Advances in neural information processing systems , 33:22268--22281, 2020

    work page 2020

  68. [68]

    Non-commutative L^p -spaces

    Gilles Pisier and Quanhua Xu. Non-commutative L^p -spaces. In Williams B. Johnson and Joram Lindenstrauss, editors, Handbook of the geometry of B anach spaces , volume 2, pages 1459--1517. Elsevier, 2003

    work page 2003

  69. [69]

    Solvable model of a spin-glass

    David Sherrington and Scott Kirkpatrick. Solvable model of a spin-glass. Physical review letters , 35(26):1792, 1975

    work page 1975

  70. [70]

    Sum-of-Squares & Gaussian Processes I: Certification

    Juspreet Singh Sandhu and Jonathan Shi. Sum-of-Squares & Gaussian Processes I: Certification. arXiv preprint arXiv:2401.14383 , 2024

    work page arXiv 2024

  71. [71]

    Free energy landscapes in spherical spin glasses

    Eliran Subag. Free energy landscapes in spherical spin glasses. arXiv preprint arXiv:1804.10576 , 2018

    work page arXiv 2018

  72. [72]

    Following the Ground States of Full-RSB Spherical Spin Glasses

    Eliran Subag. Following the Ground States of Full-RSB Spherical Spin Glasses. Communications on Pure and Applied Mathematics , 74(5):1021--1044, 2021

    work page 2021

  73. [73]

    Theory of Operator Algebras I , volume 124 of Encyclopaedia of Mathematical Sciences

    Masamichi Takesaki. Theory of Operator Algebras I , volume 124 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin Heidelberg, 2002

    work page 2002

  74. [74]

    The P arisi formula

    Michel Talagrand. The P arisi formula. Annals of mathematics , pages 221--263, 2006

    work page 2006

  75. [75]

    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

    work page 2010

  76. [76]

    Mean Field Models for Spin Glasses: Advanced replica-symmetry and low temperature

    Michel Talagrand. Mean Field Models for Spin Glasses: Advanced replica-symmetry and low temperature . Springer, 2011

    work page 2011

  77. [77]

    Dykema, and Alexandru Nica

    Dan-Virgil Voiculescu, Kenneth J. Dykema, and Alexandru Nica. Free Random Variables , volume 1 of CRM Monograph Series . American Mathematical Society, Providence, 1992

    work page 1992

  78. [78]

    Symmetries of some reduced free product C ^* -algebras

    Dan-Virgil Voiculescu. Symmetries of some reduced free product C ^* -algebras. In Huzihiro Araki, Calvin C. Moore, S erban-Valentin Stratila, and Dan-Virgil Voiculescu, editors, Operator Algebras and their Connections with Topology and Ergodic Theory , pages 556--588. Springer Berlin Heidelberg, Berlin, Heidelberg, 1985

    work page 1985

  79. [79]

    Addition of certain non-commuting random variables

    Dan-Virgil Voiculescu. Addition of certain non-commuting random variables. Journal of Functional Analysis , 66(3):323--346, 1986

    work page 1986

  80. [80]

    Limit laws for Random matrices and free products

    Dan-Virgil Voiculescu. Limit laws for Random matrices and free products. Inventiones mathematicae , 104(1):201--220, Dec 1991

    work page 1991

Showing first 80 references.