Rapid Mixing for Colorings via Spectral Independence

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

doi:10.1137/1.9781611976465.94

Cited by 33 publications

(24 citation statements)

References 32 publications

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“…Then for some absolute constant ǫ ≈ 10 −5 , if q ≥ ( 11 6 − ǫ)∆ , then the uniform distribution over proper list-colorings for (G, L) is ℓ ∞ -independent with parameter O(1). Theorem 3.6 (See proof of Theorem 9 [Che+21], see also Lemma 6.1 [Fen+21]). Let ǫ > 0, and suppose that (G, L) is a list colouring instance where G = (V, E) is a triangle-free graph of maximum degree ∆ and L = (L(v)) v∈V is a collection of color lists of maximum length q ≥ (1 + ǫ)α * ∆ + 1, α * ≈ 1.763.…”

Section: Gibbs Distributions Of Spin Systemsmentioning

confidence: 99%

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Kaufman¹,

Kyng²,

Soldà³

2021

Preprint

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We show new scalar and matrix Chernoff-style concentration bounds for a broad class of probability distributions over {0, 1} n . Building on developments in high-dimensional expanders (Kaufman and Mass ITCS'17, Dinur and Kaufman FOCS'17, Kaufman and Oppenheim Combinatorica'20) and matroid theory (Adiprasito et al. Ann. Math.'18), a breakthrough result of Anari, Liu, Oveis and Vinzant (STOC '19) showed that the up-down random walk on matroid bases has polynomial mixing time -making it possible to efficiently sample from the (weighted) uniform distribution over matroid bases. Since then, there has been a flurry of related work proving polynomial mixing times for random walks used to sample from a wide range of discrete probability distributions. Many works have observed that as a corollary of their mixing time analysis, one can obtain scalar concentration for 1-Lipschitz functions of samples from the stable distribution via standard arguments that convert bounds on the modified log-Sobolev (MLS) or Poincaré constant of a random walk into concentration results for the associated stable distribution of the random walk, see for example Hermon and Salez (arXiv'19). Several recent works have considered a matrix analog of the Poincaré inequality for a random walk with an associated stable distribution. Using this matrix Poincaré inequality, these works have derived a concentration result for matrix-valued spectral norm-Lipschitz functions of samples from the distribution, see for example Auon et al. (Adv. Math.'20). Unfortunately, these bounds are weak in many important regimes.A recently developed strategy for analyzing up-down walks is based around a novel notion of spectral independence, which quantifies the dependence between variables in a distribution over {0, 1} n using the largest eigenvalue of an associated pairwise influence matrix I, see Anari et al. (FOCS'20). Many works on spectral independence have in fact bounded a stronger quantity I ∞→∞ ≥ λ max (I), which we call ℓ ∞ -independence. We show that any distribution over {0, 1} n which has bounded ℓ ∞ -independence satisfies a matrix Chernoff bound that in key regimes is much stronger than spectral norm-Lipschitz function concentration bounds derived from matrix Poincaré inequalities. Our bounds match the matrix Chernoff bound for independent random variables due to Tropp, which is the strongest known in many cases and is essentially tight for several key settings. For spectral graph sparsification, our matrix concentration results are exponentially stronger than those obtained from matrix Poincaré inequalities. Our matrix Chernoff bound is a broad generalization and strengthening of the matrix Chernoff bound of Kyng and Song (FOCS'18). Using our bound, we can conclude as a corollary that a union of O(log |V |) random spanning trees gives a spectral graph sparsifier of a graph with |V | vertices with high probability, matching results for independent edge sampling, and matching lower bounds from Kyng and Song. This improves on the O(log 2 |V |) spanning tr...

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Section: Gibbs Distributions Of Spin Systemsmentioning

confidence: 99%

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Kaufman¹,

Kyng²,

Soldà³

2021

Preprint

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“…A distribution over {−1, +1} is said to be -spectrally independent with all fields if max (Ψ inf ( ) ) ≤ for any = ( ) ∈ ∈ R >0 . The notion of spectral independence was first introduced in [ALO20], then further developed in [CGŠV21,FGYZ21]. Remark that if a distribution is -spectrally independent in ∞-norm with all fields, then it must be -spectrally independent with all fields.…”

Section: F L C U B Fmentioning

confidence: 99%

Optimal Mixing Time for the Ising Model in the Uniqueness Regime

Chen¹,

Feng²,

Yin³

et al. 2021

Preprint

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A. We prove an optimal ( log ) mixing time of the Glauber dynamics for the Ising models with edge activity ∈ Δ−2 Δ , Δ Δ−2 . This mixing time bound holds even if the maximum degree Δ is unbounded. We refine the boosting technique developed in [CFYZ21], and prove a new boosting theorem by utilizing the entropic independence defined in [AJK + 21]. The theorem relates the modified log-Sobolev (MLS) constant of the Glauber dynamics for a near-critical Ising model to that for an Ising model in a sub-critical regime.

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“…Of direct relevance to us is the work of Anari, Liu, and Oveis Gharan [ALOG20], who introduced the notion of spectral independence (see Section 2.2 for an introduction) as a way of proving that the Glauber dynamics mixes rapidly. This notion, introduced in [ALOG20] for Boolean spin systems, was further developed in the works [FGYZ21,CGŠV21]. We defer a precise definition to Definition 2.3 in Section 2.2, but the upshot is the following:…”

Section: Introductionmentioning

confidence: 99%

“…• ([ALOG20], extended by [FGYZ21,CGŠV21]; see also Theorem 2.5 below) For a (C, η)spectrally independent q-spin system on a graph G = (V, E), the Glauber dynamics mixes in time…”

Section: Introductionmentioning

confidence: 99%

“…While this approach was successful in providing the first polynomial time approximate sampling algorithms for many interesting models, the drawback is that the parameter C can be quite a large constant; for instance, in the case of proper colorings of triangle free graphs of maximum degree ∆ with (1.763.. + δ)∆ colors, the best known bound [FGYZ21,CGŠV21] is C = O(1/δ), which leads to a mixing time of the form…”

Section: Introductionmentioning

confidence: 99%

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Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

Jain,

Pham,

Vuong

2021

Preprint

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We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent q-spin system on a graph G = (V, E) with maximum degree ∆. Notably, for several interesting examples, our bound covers the entire regime of ∆ excluded by arguments based on coupling with the stationary distribution. As concrete applications, by combining our new lower bound with known spectral independence computations and known coupling arguments:• We show that for a triangle-free graph G = (V, E) with maximum degree ∆ ≥ 3, the Glauber dynamics for the uniform distribution on proper k-colorings with k ≥ (1.763 • • • + δ)∆ colors has spectral gap Ωδ (|V | −1 ). Previously, such a result was known either if the girth of G is at least 5 [Dyer et. al, FOCS 2004], or under restrictions on ∆ [Chen et. al, STOC 2021; Hayes-Vigoda, FOCS 2003].• We show that for a regular graph G = (V, E) with degree ∆ ≥ 3 and girth at least 6, and for any ε, δ > 0, the partition function of the hardcore model with fugacity λ ≤ (1 − δ)λ c (∆) may be approximated within a (1 + ε)-multiplicative factor in time Õδ (n 2 ε −2 ). Previously, such a result was known if the girth is at least 7 [Efthymiou et. al, SICOMP 2019].• We show for the binomial random graph G(n, d/n) with d = O(1), with high probability, an approximately uniformly random matching may be sampled in time O d (n 2+o(1) ). This improves the corresponding running time of Õd (n 3 ) due to [Jerrum-Sinclair, SICOMP 1989;Jerrum, 2003].

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Rapid Mixing for Colorings via Spectral Independence

Cited by 33 publications

References 32 publications

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Scalar and Matrix Chernoff Bounds from $\ell_{\infty}$-Independence

Optimal Mixing Time for the Ising Model in the Uniqueness Regime

Spectral independence, coupling with the stationary distribution, and the spectral gap of the Glauber dynamics

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