5,345 research outputs found
Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems
We obtain new principles for transferring log-Sobolev and Spectral-Gap
inequalities from a source metric-measure space to a target one, when the
curvature of the target space is bounded from below. As our main application,
we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of
various conservative spin system models, consisting of non-interacting and
weakly-interacting particles, constrained to conserve the mean-spin. When the
self-interaction is a perturbation of a strongly convex potential, this
partially recovers and partially extends previous results of Caputo,
Chafa\"{\i}, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg
and Yau. When the self-interaction is only assumed to be (non-strongly) convex,
as in the case of the two-sided exponential measure, we obtain sharp estimates
on the system's spectral-gap as a function of the mean-spin, independently of
the size of the system.Comment: 57 page
Dynamic temperature selection for parallel-tempering in Markov chain Monte Carlo simulations
Modern problems in astronomical Bayesian inference require efficient methods
for sampling from complex, high-dimensional, often multi-modal probability
distributions. Most popular methods, such as Markov chain Monte Carlo sampling,
perform poorly on strongly multi-modal probability distributions, rarely
jumping between modes or settling on just one mode without finding others.
Parallel tempering addresses this problem by sampling simultaneously with
separate Markov chains from tempered versions of the target distribution with
reduced contrast levels. Gaps between modes can be traversed at higher
temperatures, while individual modes can be efficiently explored at lower
temperatures. In this paper, we investigate how one might choose the ladder of
temperatures to achieve more efficient sampling, as measured by the
autocorrelation time of the sampler. In particular, we present a simple,
easily-implemented algorithm for dynamically adapting the temperature
configuration of a sampler while sampling. This algorithm dynamically adjusts
the temperature spacing to achieve a uniform rate of exchanges between chains
at neighbouring temperatures. We compare the algorithm to conventional
geometric temperature configurations on a number of test distributions and on
an astrophysical inference problem, reporting efficiency gains by a factor of
1.2-2.5 over a well-chosen geometric temperature configuration and by a factor
of 1.5-5 over a poorly chosen configuration. On all of these problems a sampler
using the dynamical adaptations to achieve uniform acceptance ratios between
neighbouring chains outperforms one that does not.Comment: 21 pages, 21 figure
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