Rapid Mixing for Colorings via Spectral Independence

Chen, Zongchen; Galanis, Andreas; Štefankovič, Daniel; Vigoda, Eric

doi:10.48550/arxiv.2007.08058

Cited by 3 publications

(15 citation statements)

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“…and the spectral independence correspondingly. We remark that Definition 1.8 is weaker than the notion of spectral independence given in [Fen+20], and for all current applications as in [Che+20;Fen+20] or here in this paper, both definitions work.…”

Section: Results For General Spin Systemsmentioning

confidence: 98%

“…The second condition is the notion of spectral independence, first given by [ALO20] and later generalized to multi-spin systems in [Che+20;Fen+20]. Here we use the definitions from [Che+20].…”

Section: Results For General Spin Systemsmentioning

confidence: 99%

“…The marginal boundedness is a trivial bound. The spectral independence was previously established for antiferromagnetic 2-spin systems including the hard-core model and the Ising model in the whole uniqueness region [ALO20; CLV20], and for random q-colorings when q is sufficiently large [Che+20;Fen+20]. For the monomer-dimer model, spectral independence is not known previously.…”

Section: Wrapping Upmentioning

confidence: 97%

“…Our approach holds for general spin systems. The spectral independence approach, first introduced for 2-spins in [ALO20] and subsequently extended to q-spins in [Che+20;Fen+20], considers the qn×qn influence matrix. For spins i, j ∈ [q] and vertices u, v ∈ V , the entry ((u, i), (v, j)) of the influence matrix measures the effect of vertex u having spin i on the marginal probability that vertex v has spin j, see Definition 1.7 for a precise statement.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion

Chen

Liu

Vigoda

2020

Preprint

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We prove an optimal mixing time bound on the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari et al. ( 2020) and shows O(n log n) mixing time on any n-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hard-core model on independent sets weighted by a fugacity λ, we establish O(n log n) mixing time for the Glauber dynamics on any n-vertex graph of constant maximum degree ∆ when λ < λ c (∆) where λ c (∆) is the critical point for the uniqueness/non-uniqueness phase transition on the ∆-regular tree. More generally, for any antiferromagnetic 2-spin system we prove O(n log n) mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain O(n log n) mixing for q-colorings of triangle-free graphs of maximum degree ∆ when the number of colors satisfies q > α∆ where α ≈ 1.763, and O(m log n) mixing for generating random matchings of any graph with bounded degree and m edges.Our approach is based on two steps. First, we show that the approximate tensorization of entropy (i.e., factorizing entropy into single vertices), which is a key step for establishing the modified log-Sobolev inequality in many previous works, can be deduced from entropy factorization into blocks of fixed linear size. Second, we adapt the local-to-global scheme of Alev and Lau (2020) to establish such block factorization of entropy in a more general setting of pure weighted simplicial complexes satisfying local spectral expansion; this also substantially generalizes the result of Cryan et al. (2019).

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Section: Results For General Spin Systemsmentioning

confidence: 98%

Section: Results For General Spin Systemsmentioning

confidence: 99%

Section: Wrapping Upmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion

Chen

Liu

Vigoda

2020

Preprint

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“…A well-known result due to [Jer95] using path coupling shows that if |L(v)| > 2∆ for all v ∈ V , then there is a contractive one-step coupling for the Glauber dynamics which yields O(n log n) mixing. As noted in [CLV21], one can adapt the argument of [GKM15] to obtain strong spatial mixing when |L(v)| > 2∆, and use the arguments of [Che+20;Fen+20] to deduce spectral independence in this regime. However, it is still open whether one can obtain strong spatial mixing below the 2∆ threshold; see [GKM15;Eft+19] for results going below 2∆ on special classes of graphs.…”

Section: Spectral Independence For Proper List-coloringsmentioning

confidence: 99%

From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk

Liu¹

2021

Preprint

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We show that the existence of a "good" coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-toglobal arguments [CLV21], we establish asymptotically optimal lower bounds on the standard and modified log-Sobolev constants for the Glauber dynamics for sampling from spin systems on bounded-degree graphs when a curvature condition [Oll09] is satisfied. To achieve this, we use Stein's method for Markov chains [BN19; RR19] to show that a "good" coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of [ALO20].Our primary application is to sampling proper list-colorings on bounded-degree graphs. In particular, combining the coupling for the flip dynamics given by [Vig00; Che+19] with our techniques, we show optimal O(n log n) mixing for the Glauber dynamics for sampling proper list-colorings on any boundeddegree graph with maximum degree ∆ whenever the size of the color lists are at least 11 6 − ǫ ∆, where ǫ ≈ 10 −5 is small constant. While O(n 2 ) mixing was already known before, our approach additionally yields Chernoff-type concentration bounds for Hamming Lipschitz functions in this regime, which was not known before. Our approach is markedly different from prior works establishing spectral independence for spin systems using spatial mixing [ALO20; CLV20; Che+20; Fen+20], which crucially is still open in this regime for proper list-colorings.

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Rapid Mixing for Colorings via Spectral Independence

Chen¹,

Galanis²,

Štefankovič³

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted independent sets. We develop the spectral independence approach for colorings, and obtain new algorithmic results for the corresponding counting/sampling problems.Let α * ≈ 1.763 denote the solution to exp(1/x) = x and let α > α * . We prove that, for any triangle-free graph G = (V, E) with maximum degree ∆, for all q ≥ α∆ + 1, the mixing time of the Glauber dynamics for qcolorings is polynomial in n = |V |, with the exponent of the polynomial independent of ∆ and q. In comparison, previous approximate counting results for colorings held for a similar range of q (asymptotically in ∆) but with larger girth requirement or with a running time where the polynomial exponent depended on ∆ and q (exponentially). One further feature of using the spectral independence approach to study colorings is that it avoids many of the technical complications in previous approaches caused by coupling arguments or by passing to the complex plane; the key improvement on the running time is based on relatively simple combinatorial arguments which are then translated into spectral bounds.

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Rapid Mixing for Colorings via Spectral Independence

Cited by 3 publications

References 40 publications

Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion

Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion

From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk

Rapid Mixing for Colorings via Spectral Independence

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