Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

Anari, Nima; Liu, Kuikui; Gharan, Shayan Oveis

doi:10.48550/arxiv.2001.00303

Cited by 9 publications

(54 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our work builds upon the spectral independence approach introduced by Anari, Liu, and Oveis Gharan [2], which in turn utilizes the high-dimensional expander work of Alev and Lau [1]. Consider a graph G = (V, E) of maximum degree ∆.…”

Section: Proof Approachmentioning

confidence: 99%

“…To this vein, MCMC methods typically give much faster (randomized) algorithms, however correspond-ing results were lacking until a recent breakthrough result of Anari, Liu and Oveis Gharan [2], who proved rapid mixing of the Glauber dynamics for the hard-core model, matching the parameter range of the aforementioned non-MCMC approaches and also improving the running time with a polynomial exponent which is independent of the degree bound ∆. They introduced a spectral independence approach which utilizes high-dimensional expander results of Alev and Lau [1] (cf.…”

Section: Introductionmentioning

confidence: 99%

“…The work of [2] establishes that, for 2-spin systems, it suffices to bound the largest eigenvalue of the n × n influence matrix I where the (v, w) entry captures the influence of the fixed spin at vertex v on the marginal probability at vertex w; we explain this in more detail in Section 1.1. The running time of the result of [2] was further improved in [5], who also generalised the approach to antiferromagnetic 2-spin systems up to the tree-uniqueness threshold by showing how to utilize potential-function arguments that were previously used to establish SSM. Going beyond 2-spin systems, all of these methods become harder to control even well above the treeuniqueness threshold, q = ∆ + 1, which marks the onset of computational hardness (even for triangle-free graphs, see [8]).…”

Section: Introductionmentioning

confidence: 99%

“…Our main contribution is to develop the spectral independence approach of [1,2] for colorings, and analyze Glauber dynamics in the regime q ≥ α∆ + 1 for all α > α * on triangle-free graphs. Our result applies for all ∆ and we show that the exponent of the mixing time does not depend on ∆ and q, yielding substantially faster randomized algorithms for sampling/counting colorings than the previous deterministic ones (at the expense of using randomness).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Rapid Mixing for Colorings via Spectral Independence

Chen

Galanis

Štefankovič

et al. 2020

Preprint

View full text Add to dashboard Cite

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 hardcore 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 trianglefree graph G = (V, E) with maximum degree ∆, for all q ≥ α∆ + 1, the mixing time of the Glauber dynamics for q-colorings 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.

show abstract

Section: Proof Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Rapid Mixing for Colorings via Spectral Independence

Chen

Galanis

Štefankovič

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…HDX are a family of expanding complexes that have seen an explosion of work in recent years, leading to major breakthroughs across a number of areas including (among others) the recent construction of c3-LTCs [DEL + 21], and efficient approximate sampling for many important systems (e.g. for matroid bases [ALOV19], independent sets [ALO20], Ising models [AJK + 21], and more). Our results lead to a new understanding of the structure of boolean functions on HDX, including a tight analog of the KKL Theorem, and a characterization of nonexpanding sets similar to that used in the proof of 2-2 Games [KMS18].…”

Section: Introductionmentioning

confidence: 99%

Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments

Bafna¹,

Hopkins²,

Kaufman³

et al. 2021

Preprint

View full text Add to dashboard Cite

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the p-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets.Unlike previous settings satisfying hypercontractivity, HDX can be asymmetric, sparse, and very far from products, which makes the application of traditional proof techniques challenging. We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global method for analyzing higher moments. Interestingly, unlike analogous second moment methods that apply equally across all types of expanding complexes, our tools rely inherently on simplicial structure. This suggests a new distinction among high dimensional expanders based upon their behavior beyond the second moment.

show abstract

Rapid Mixing for Colorings via Spectral Independence

Chen¹,

Galanis²,

Štefankovič³

et al. 2021

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

View full text Add to dashboard Cite

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.

show abstract

Spectral Independence in High-Dimensional Expanders and Applications to the Hardcore Model

Cited by 9 publications

References 31 publications

Rapid Mixing for Colorings via Spectral Independence

Rapid Mixing for Colorings via Spectral Independence

Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments

Rapid Mixing for Colorings via Spectral Independence

Contact Info

Product

Resources

About