Gibbs random fields with multiple pairwise pixel interactionshave good potentialities in modeling natural image texturesbecause allow for learning both the structure and strengths ofpixel interactions from a given training sample. The learningscheme is based on the maximum likelihood estimate (MLE) ofGibbs potentials that specify the interaction strenghts. Thisscheme is amplified here by deduci...

The cosmological constant is not an absolute constant. The gravitating part of the vacuum energy is adjusted to the energy density of matter and to other types of the perturbations of the vacuum. We discuss how the vacuum energy responds (i) to the curvature of space in the Einstein closed Universe; (ii) to the expansion rate in the de Sitter Universe; and (iii) to the rotation in the Gödel Uni...

The cosmological constant is not an absolute constant. The gravitating part of the vacuum energy is adjusted to the energy density of matter and to other types of the perturbations of the vacuum. We discuss how the vacuum energy responds (i) to the curvature of space in the Einstein closed Universe; (ii) to the expansion rate in the de Sitter Universe; and (iii) to the rotation in the Gödel Uni...

Traditional derivation of Gibbs canonical distribution and the justification of thermodynamics are based on the assumption concerning an isoenergetic ergodicity of a system of n weakly interacting identical subsystems and passage to the limit n → ∞. In the presented work we develop another approach to these problems assuming that n is fixed and n ≥ 2. The ergodic hypothesis (which frequently is...

Stein’s method provides a way of bounding the distance of a probability distribution to a target distribution μ. Here we develop Stein’s method for the class of discrete Gibbs measures with a density e , where V is the energy function. Using size bias couplings, we treat an example of Gibbs convergence for strongly correlated random variables due to Chayes and Klein [Helv. Phys. Acta 67 (1994) ...

This project applies Gibbs Sampling based on different Markov Random Fields (MRF) structures to solve the image denoising problem. Compared with methods like gradient ascent, one important advantage that Gibbs Sampling has is that it provides balances between exploration and exploitation. This project also tested behaviors of different MRF structures, such as the high-order MRFs, various loss f...

Optimization based on k-step contrastive divergence (CD) has become a common way to train restricted Boltzmann machines (RBMs). The k-step CD is a biased estimator of the log-likelihood gradient relying on Gibbs sampling. We derive a new upper bound for this bias. Its magnitude depends on k, the number of variables in the RBM, and the maximum change in energy that can be produced by changing a ...

A new approach to align an image of a medical object with a given prototype is proposed. Visual appearance of the images, after equalizing their signals, is modelled with a new Markov-Gibbs random field with pairwise interaction model. Similarity to the prototype is measured by a Gibbs energy of signal co-occurrences in a characteristic subset of pixel pairs derived automatically from the proto...

where z denotes the microscopic configuration and ρeq(z)dz is the equilibrium measure for the microscopic dynamics. In some cases the actual Hamiltonian (total energy) may be expressible in terms of the coarse-grained variables and is strictly conserved, but this is not always the case since some of the energy may in fact be incorporated in the coarse-grained free energy as an “entropy”. The eq...

A Gibbs-like approach for simultaneous multi-scale correlation functions in random, time-dependent, multiplicative processes for the turbulent energy cascade is investigated. We study the optimal log-normal Gibbs-like distribution able to describe the subtle effects induced by non-trivial time dependency on both single-scale (structure functions) and multi-scale correlation functions. We provid...