# Modified log-Sobolev inequalities for strong-Rayleigh measures

@article{Hermon2019ModifiedLI, title={Modified log-Sobolev inequalities for strong-Rayleigh measures}, author={Jonathan Hermon and Justin Salez}, journal={arXiv: Probability}, year={2019} }

We establish universal modified log-Sobolev inequalities for reversible Markov chains on the boolean lattice $\{0,1\}^n$, under the only assumption that the invariant law $\pi$ satisfies a form of negative dependence known as the \emph{stochastic covering property}. This condition is strictly weaker than the strong Rayleigh property, and is satisfied in particular by all determinantal measures, as well as any product measure over the set of bases of a balanced matroid. In the special case where…

## 28 Citations

Entropic Independence II: Optimal Sampling and Concentration via Restricted Modified Log-Sobolev Inequalities

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A framework for obtaining tight mixing time bounds for Markov chains based on restricted modified log-Sobolev inequalities is introduced, and the perhaps surprising result that one can sample from tree-unique hardcore and Ising models in time Õδ(n), without even having enough time to read all edges of the graph is obtained.

Entropy decay in the Swendsen-Wang dynamics on ${\mathbb Z}^d$

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We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ${\mathbb Z}^d$. This dynamics is a widely used Markov chain that has…

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A Matrix Bernstein Inequality for Strong Rayleigh Distributions.

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The Entropy method provides a powerful framework for proving scalar concentration inequalities by establishing functional inequalities like Poincare and log-Sobolev inequalities. These inequalities…

Isotropy and Log-Concave Polynomials: Accelerated Sampling and High-Precision Counting of Matroid Bases

- Computer Science, Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
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A new approximate sampling algorithm that leverages isotropy for the class of distributions that have a log-concave generating polynomial that includes determinantal point processes, strongly Rayleigh distributions, and uniform distributions over matroid bases is designed.

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The results show that as datasets grow, Gaussian process posteriors can truly be approximated cheaply, and provide a concrete rule for how to increase $M$ in continual learning scenarios.

Entropy dissipation estimates for inhomogeneous zero-range processes

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Introduced by Lu and Yau (CMP, 1993), the martingale decomposition method is a powerful recursive strategy that has produced sharp log-Sobolev inequalities for homogeneous particle systems. However,…

Modified log-Sobolev Inequalities for Strongly Log-Concave Distributions

- Mathematics, Computer Science2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
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We show that the modified log-Sobolev constant for a natural Markov chain which converges to an r-homogeneous strongly log-concave distribution is at least 1/r. Applications include an asymptotically…

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Introduced by Lu and Yau (CMP, 1993), the martingale decomposition method is a powerful recursive strategy that has produced sharp log-Sobolev inequalities for homogeneous particle systems. However,…

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We show that the modified log-Sobolev constant for a natural Markov chain which converges to an r-homogeneous strongly log-concave distribution is at least 1/r. Applications include an asymptotically…

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