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1. Skinny Gibbs: A Consistent and Scalable Gibbs Sampler for Model Selection

   https://doi.org/10.1080/01621459.2018.1482754  

   Narisetty, Naveen N.; Shen, Juan; He, Xuming (June 2018, Journal of the American Statistical Association)
2. HOGWILD!-Gibbs can be PanAccurate

   Daskalakis, Constantinos; Dikkala, Nishanth; Jayanti, Siddhartha (January 2018, 32nd Annual Conference on Neural Information Processing Systems)

   Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by O(τlogn), where n is the number of variables in the graphical model, and τ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in n. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the true model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings. 

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3. HOGWILD!-Gibbs can be PanAccurate

   Daskalakis, Constantinos; Dikkala, Nishanth; Jayanti, Siddhartha (January 2018, Journal of neural information processing)

   Asynchronous Gibbs sampling has been recently shown to be fast-mixing and an accurate method for estimating probabilities of events on a small number of variables of a graphical model satisfying Dobrushin's condition~\cite{DeSaOR16}. We investigate whether it can be used to accurately estimate expectations of functions of {\em all the variables} of the model. Under the same condition, we show that the synchronous (sequential) and asynchronous Gibbs samplers can be coupled so that the expected Hamming distance between their (multivariate) samples remains bounded by O(τlogn), where n is the number of variables in the graphical model, and τ is a measure of the asynchronicity. A similar bound holds for any constant power of the Hamming distance. Hence, the expectation of any function that is Lipschitz with respect to a power of the Hamming distance, can be estimated with a bias that grows logarithmically in n. Going beyond Lipschitz functions, we consider the bias arising from asynchronicity in estimating the expectation of polynomial functions of all variables in the model. Using recent concentration of measure results, we show that the bias introduced by the asynchronicity is of smaller order than the standard deviation of the function value already present in the true model. We perform experiments on a multi-processor machine to empirically illustrate our theoretical findings. 

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4. On local quantum Gibbs states

   https://doi.org/10.1063/5.0058574  

   Duboscq, Romain; Pinaud, Olivier (October 2022, Journal of Mathematical Physics)

   We address in this work the problem of minimizing quantum entropies under local constraints. We suppose that macroscopic quantities, such as the particle density, current, and kinetic energy, are fixed at each point of Rd and look for a density operator over L2(Rd), minimizing an entropy functional. Such minimizers are referred to as local Gibbs states. This setting is in contrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises, for instance, in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy. 

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5. Uniqueness of Gibbs measures for continuous hardcore models

   https://doi.org/10.1214/18-AOP1298  

   Gamarnik, David; Ramanan, Kavita (July 2019, The Annals of Probability)

   null (Ed.)