The Worm Process for the Ising Model is Rapidly Mixing

Collevecchio, Andrea; Garoni, Timothy M.; Hyndman, Timothy; Tokarev, Daniel

doi:10.1007/s10955-016-1572-2

Cited by 17 publications

(21 citation statements)

References 71 publications

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“…We further discuss the real cases 𝜆 = ±1 which are not explicitly covered by Theorem 1. The case 𝜆 = 1 admits an FPRAS [26,22,11], but the existence of a deterministic approximation scheme is open. We study the case 𝜆 = −1 in more detail in Section 9, where we show that the problem is not #P-hard (assuming #P ≠ NP): using the 'high-temperature' expansion of the model, we show an oddsubgraphs formulation of the partition function (Lemma 39), which is then used to conclude (Theorem 40) that the sign of the partition function can be determined trivially, while the problem of approximating the norm of the partition function for all Δ ≥ 3 is equivalent to the problem of approximately counting the number of perfect matchings (even on unbounded-degree graphs); the complexity of the latter is an open problem in general, but it can be approximated with an NP-oracle [27], therefore precluding #P-hardness.…”

Section: Our Resultsmentioning

confidence: 99%

“…The Lee-Yang theorem was recently used by Liu, Sinclair and Srivastava [35] to obtain an FPTAS for approximating the partition function for values 𝜆 ∈ C that do not lie on the unit circle. This result can be viewed as a derandomisation of the Markov chain-based randomised algorithm by Jerrum and Sinclair [26] for 𝜆 > 0 (see also [22,11]), solving a longstanding problem. 2 As noted in [35,Remark p. 290], the 'no-field' case |𝜆| = 1 is unclear, since, on the one hand, we have the algorithm by [26] for 𝜆 = 1 and, on the other hand, it is known that Lee-Yang zeros are dense on the unit circle.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

Buys

Galanis

Patel

et al. 2022

Forum of Mathematics, Sigma

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We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda $ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics but have also been key in the recent deterministic approximation scheme for all $|\lambda |\neq 1$ by Liu, Sinclair and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle and on the tantalising question of what happens around $\lambda =1$ , where, on one hand, the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability and, on the other hand, the presence of Lee–Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda =1$ and, more generally, on the entire unit circle. For an integer $\Delta \geq 3$ and edge interaction parameter $b\in (0,1)$ , we show $\mathsf {\#P}$ -hardness for approximating the partition function on graphs of maximum degree $\Delta $ on the arc of the unit circle where the Lee–Yang zeros are dense. This result contrasts with known approximation algorithms when $|\lambda |\neq 1$ or when $\lambda $ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee–Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.

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Section: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

Buys

Galanis

Patel

et al. 2022

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

show abstract

“…Further evidence for the hardness of this problem comes from the fact that sampling is hard in the antiferromagnetic setting [56,30] and in the ferromagnetic model in the presence of inconsistent magnetic fields [33] (i.e., the vertex potential of distinct vertices may have different signs). In summary, the algorithmic results of [19] are most interesting for the ferromagnetic Ising model (with consistent fields), where there are known polynomial running time algorithms for estimating the pairwise covariances (see, e.g., [42,48,37,17]).…”

Section: Lower Bounds For the Ising Modelmentioning

confidence: 99%

Lower bounds for testing graphical models: colorings and antiferromagnetic Ising models

Bezáková¹,

Blanca²,

Chen³

et al. 2019

Preprint

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“…The proof of the above lemma is given in Section B.1. Collevecchio et al [18] recently proved that the worm algorithm mixes in polynomial time when the weights are uniform, i.e., equal. We extend the result to our case of non-uniform weights.…”

Section: Appendix B Proof Of Theoremmentioning

confidence: 99%

“…In the proof, we first show that MC induced by the Worm algorithm mixes in polynomial time, and then prove that acceptance of a 2-regular loop, i.e., line 6 of Algorithm 1, occurs with high probability. Notice that the uniform-weight version of the former proof, i.e., fast mixing, was recently proven in [18]. For completeness of the material exposition, we present the general case proof of interest for us.…”

mentioning

confidence: 91%

Sythesis of MCMC and Belief Propagation

Ahn

Chertkov

Shin

2016

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Markov Chain Monte Carlo (MCMC) and Belief Propagation (BP) are the most popular algorithms for computational inference in Graphical Models (GM). In principle, MCMC is an exact probabilistic method which, however, often suffers from exponentially slow mixing. In contrast, BP is a deterministic method, which is typically fast, empirically very successful, however in general lacking control of accuracy over loopy graphs. In this paper, we introduce MCMC algorithms correcting the approximation error of BP, i.e., we provide a way to compensate for BP errors via a consecutive BP-aware MCMC. Our framework is based on the Loop Calculus approach which allows to express the BP error as a sum of weighted generalized loops. Although the full series is computationally intractable, it is known that a truncated series, summing up all 2-regular loops, is computable in polynomial-time for planar pair-wise binary GMs and it also provides a highly accurate approximation empirically. Motivated by this, we first propose a polynomial-time approximation MCMC scheme for the truncated series of general (non-planar) pair-wise binary models. Our main idea here is to use the Worm algorithm, known to provide fast mixing in other (related) problems, and then design an appropriate rejection scheme to sample 2-regular loops. Furthermore, we also design an efficient rejection-free MCMC scheme for approximating the full series. The main novelty underlying our design is in utilizing the concept of cycle basis, which provides an efficient decomposition of the generalized loops. In essence, the proposed MCMC schemes run on transformed GM built upon the non-trivial BP solution, and our experiments show that this synthesis of BP and MCMC outperforms both direct MCMC and bare BP schemes.

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The Worm Process for the Ising Model is Rapidly Mixing

Cited by 17 publications

References 71 publications

Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

Lower bounds for testing graphical models: colorings and antiferromagnetic Ising models

Sythesis of MCMC and Belief Propagation

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