A tight degree 4 sum-of-squares lower bound for the Sherrington–Kirkpatrick Hamiltonian

Kunisky, Dmitriy; Bandeira, Afonso S.

doi:10.1007/s10107-020-01558-2

Cited by 7 publications

(7 citation statements)

References 29 publications

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“…To complete that computation, we will show moreover that Y (n) and the associated pseudoexpectation E arise from a particular case of the spectral extension construction proposed by [KB20,Kun20] which describes Y (n) as a Gram matrix of certain polynomials under the apolar inner product. This observation will allow us to rephrase the computation of the eigenvalues Y (n) in terms of this Hilbert space of polynomials, which we can carry out in closed form.…”

Section: Main Results and Proof Ideasmentioning

confidence: 99%

“…Using these, we prove and elaborate on Laurent's observations.Our arguments have two features that may be of independent interest. First, we show that the Grigoriev-Laurent pseudomoments are a special case of a Gram matrix construction of pseudomoments proposed by Bandeira and Kunisky (2020). Second, we find a new realization of the irreducible representations of the symmetric group corresponding to Young diagrams with two rows, as spaces of multivariate polynomials that are multiharmonic with respect to an equilateral simplex.…”

mentioning

confidence: 82%

“…In such a setting, [KB20,Kun20] proposed the technique of spectral extension for determining the values of the higher-degree values of a pseudoexpectation E. Spectral extension proposes building E in a "Gramian" fashion, having E[x S x T ] = v S , v T for suitable vectors v S . Actually, these vectors may be interpreted as polynomials endowed with a particular inner product, which we review below.…”

Section: Spectral Extensions Of Pseudomomentsmentioning

confidence: 99%

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The spectrum of the Grigoriev-Laurent pseudomoments

Kunisky¹,

Moore²

2022

Preprint

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and Laurent (2003) independently showed that the sum-of-squares hierarchy of semidefinite programs does not exactly represent the hypercube {±1} n until degree at least n of the hierarchy. Laurent also observed that the pseudomoment matrices her proof constructs appear to have surprisingly simple and recursively structured spectra as n increases. While several new proofs of the Grigoriev-Laurent lower bound have since appeared, Laurent's observations have remained unproved. We give yet another, representation-theoretic proof of the lower bound, which also yields exact formulae for the eigenvalues of the Grigoriev-Laurent pseudomoments. Using these, we prove and elaborate on Laurent's observations.Our arguments have two features that may be of independent interest. First, we show that the Grigoriev-Laurent pseudomoments are a special case of a Gram matrix construction of pseudomoments proposed by Bandeira and Kunisky (2020). Second, we find a new realization of the irreducible representations of the symmetric group corresponding to Young diagrams with two rows, as spaces of multivariate polynomials that are multiharmonic with respect to an equilateral simplex.

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Section: Main Results and Proof Ideasmentioning

confidence: 99%

mentioning

confidence: 82%

Section: Spectral Extensions Of Pseudomomentsmentioning

confidence: 99%

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The spectrum of the Grigoriev-Laurent pseudomoments

Kunisky¹,

Moore²

2022

Preprint

Self Cite

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“…To the best of our knowledge, a variation of [47] for the diluted model (p < 1) has not yet been discussed. Furthermore, there is an integrality gap for the SDP relaxation of this problem in the Gaussian setting [40,48] whenever λ = 0. As we discuss in more detail in Appendix B, this implies that the value of the problem in the case τ ij = 1 converges to the largest eigenvalue of A in the limit n → ∞.…”

Section: Quadratic Quantum Speedups and Classical Speedups For Generic Instances And Spin Glassesmentioning

confidence: 99%

“…The following random instances of the MaxQP have received significant attention in recent literature [48,40,31]: we define the Gaussian orthogonal ensemble (GOE) to be the random matrix distribution over symmetric n × n matrices A with i.i.d. normal entries A ii ∼ N (0, 2/n) on the diagonal and A ij ∼ N (0, 1/n) for i < j.…”

Section: A Norms Of Random Matricesmentioning

confidence: 99%

Faster quantum and classical SDP approximations for quadratic binary optimization

Brandão

Kueng

França

2022

Quantum

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We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. Such instances include approximating the ground state of spin glasses and MaxCut on Erdös-Rényi graphs. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into approximations of the original quadratic optimization problem.

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Disordered systems insights on computational hardness

2022

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In this review article we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked tensor model and the generalized linear model. We discuss the random (non-planted) versions of these problems as prototypical optimization problems, as well as the planted versions (with a hidden solution) as prototypical problems in statistical inference and learning. Based on ideas from physics, many of these problems have transitions where they are believed to jump from easy (solvable in polynomial time) to hard (requiring exponential time). We discuss several emerging ideas in theoretical computer science and statistics that provide rigorous evidence for hardness by proving that large classes of algorithms fail in the conjectured hard regime. This includes the overlap gap property, a particular mathematization of clustering or dynamical symmetry-breaking, which can be used to show that many algorithms that are local or robust to changes in their input fail. We also discuss the sum-of-squares hierarchy, which places bounds on proofs or algorithms that use low-degree polynomials such as standard spectral methods and semidefinite relaxations, including the Sherrington–Kirkpatrick model. Throughout the manuscript we present connections to the physics of disordered systems and associated replica symmetry breaking properties.

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A tight degree 4 sum-of-squares lower bound for the Sherrington–Kirkpatrick Hamiltonian

Cited by 7 publications

References 29 publications

The spectrum of the Grigoriev-Laurent pseudomoments

The spectrum of the Grigoriev-Laurent pseudomoments

Faster quantum and classical SDP approximations for quadratic binary optimization

Disordered systems insights on computational hardness

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