Spectral Independence via Stability and Applications to Holant-Type Problems

Chen, Zongchen; Liu, Kuikui; Vigoda, Eric

doi:10.48550/arxiv.2106.03366

Cited by 7 publications

(14 citation statements)

References 35 publications

(79 reference statements)

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“…Chen et al [CLV21b] generalize the results of Alimohammadi et al [Ali+21] which formalize a connection between stability of polynomials and spectral independence of certain probability distributions. By inspection of the proof, we notice that in order to prove spectral independence, in both papers, the authors show a stronger result, namely that the distributions of interest are ℓ ∞ -independent.…”

Section: Stable Distributionsmentioning

confidence: 59%

“…Note that following the notation in [CLV21b], the Gibbs distribution is not defined on {0, 1} n but we can encode a configuration σ ∈ Q V with a binary vector ξ ∈ {0, 1} |V |q such that ξ(v, i) = 1 ⇐⇒ σ(v) = i, for all v ∈ V and i ∈ [q]. Note that the partition function is not changed by this change of encoding.…”

Section: Stable Distributionsmentioning

confidence: 99%

“…Theorem 3.10 (From the proof of Theorem 8 [CLV21b]). Let λ * ∈ R + and let Γ ⊂ C be a nonempty open connected region such that (0, λ * ) ⊆ Γ (respectively, (λ * , ∞) ⊆ Γ).…”

Section: Stable Distributionsmentioning

confidence: 99%

“…From proof of Theorem 7[CLV21b]). Let Γ ⊂ C be a non-empty open connected region such that Γ is unbounded and 0 belongs to the closure of Γ.…”

mentioning

confidence: 98%

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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: Stable Distributionsmentioning

confidence: 59%

Section: Stable Distributionsmentioning

confidence: 99%

“…Theorem 3.10 (From the proof of Theorem 8 [CLV21b]). Let λ * ∈ R + and let Γ ⊂ C be a nonempty open connected region such that (0, λ * ) ⊆ Γ (respectively, (λ * , ∞) ⊆ Γ).…”

Section: Stable Distributionsmentioning

confidence: 99%

“…From proof of Theorem 7[CLV21b]). Let Γ ⊂ C be a non-empty open connected region such that Γ is unbounded and 0 belongs to the closure of Γ.…”

mentioning

confidence: 98%

See 2 more Smart Citations

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

Kaufman¹,

Kyng²,

Soldà³

2021

Preprint

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show abstract

“…The study of the locus of complex zeros has a rich history in statistical physics, for example, in the famous Lee-Yang theorem [LY52]. In computer science, the absence of complex zeros may imply efficient approximation algorithms for partition functions [Bar17b, PR17, Reg18, HPR20, LSS19b, LSS19a, PR19, CDK + 20, GLL20, GLLZ20, HMS20, SS20, BGPR21,CLV21b]. This line of research was initiated by Barvinok's pioneering works [Bar15, Bar16, Bar17a, Bar17b, Bar17c], which used truncated Taylor expansions to approximate non-vanishing polynomials and established quasipolynomial time approximations of partition functions with no complex zeros within a region.…”

mentioning

confidence: 99%

Polynomial-Time Approximation of Zero-Free Partition Functions

Yao¹,

Yin²,

Zhang³

2022

Preprint

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A. Zero-free based algorithm is a major technique for deterministic approximate counting. In Barvinok's original framework [Bar17b], by calculating truncated Taylor expansions, a quasi-polynomial time algorithm was given for estimating zero-free partition functions. Patel and Regts [PR17] later gave a refinement of Barvinok's framework, which gave a polynomial-time algorithm for a class of zero-free graph polynomials that can be expressed as counting induced subgraphs in bounded-degree graphs.In this paper, we give a polynomial-time algorithm for estimating classical and quantum partition functions specified by local Hamiltonians with bounded maximum degree, assuming a zero-free property for the temperature. Consequently, when the inverse temperature is close enough to zero by a constant gap, we have polynomial-time approximation algorithm for all such partition functions. Our result is based on a new abstract framework that extends and generalizes the approach of Patel and Regts.

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Smoothed counting of 0–1 points in polyhedra

Barvinok

2022

Random Struct Algorithms

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Given a system of linear equations ℓifalse(xfalse)=βi$$ {\ell}_i(x)={\beta}_i $$ in an n$$ n $$‐vector x$$ x $$ of 0–1 variables, we compute the expectation of exp{}prefix−∑iγi()ℓifalse(xfalse)prefix−βi2$$ \exp \left\{-{\sum}_i{\gamma}_i{\left({\ell}_i(x)-{\beta}_i\right)}^2\right\} $$, where x$$ x $$ is a vector of independent Bernoulli random variables and γi>0$$ {\gamma}_i>0 $$ are constants. The algorithm runs in quasi‐polynomial nOfalse(lnnfalse)$$ {n}^{O\left(\ln n\right)} $$ time under some sparseness condition on the matrix of the system. The result is based on the absence of the zeros of the analytic continuation of the expectation for complex probabilities, which can also be interpreted as the absence of a phase transition in the Ising model with a sufficiently strong external field. We discuss applications to perfect matchings in hypergraphs and randomized rounding in discrete optimization.

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Spectral Independence via Stability and Applications to Holant-Type Problems

Cited by 7 publications

References 35 publications

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

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

Polynomial-Time Approximation of Zero-Free Partition Functions

Smoothed counting of 0–1 points in polyhedra

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