Bayesian optimization for computationally extensive probability distributions

An efficient method for finding a better maximizer of computationally extensive probability distributions is proposed on the basis of a Bayesian optimization technique. A key idea of the proposed method is to use extreme values of acquisition functions by Gaussian processes for the next training phase, which should be located near a local maximum or a global maximum of the probability distribution. Our Bayesian optimization technique is applied to the posterior distribution in the effective physical model estimation, which is a computationally extensive probability distribution. Even when the number of sampling points on the posterior distributions is fixed to be small, the Bayesian optimization provides a better maximizer of the posterior distributions in comparison to those by the random search method, the steepest descent method, or the Monte Carlo method. Furthermore, the Bayesian optimization improves the results efficiently by combining the steepest descent method and thus it is a powerful tool to search for a better maximizer of computationally extensive probability distributions.Comment: 13 pages, 5 figure

A Method to Change Phase Transition Nature -- Toward Annealing Method --

In this paper, we review a way to change nature of phase transition with annealing methods in mind. Annealing methods are regarded as a general technique to solve optimization problems efficiently. In annealing methods, we introduce a controllable parameter which represents a kind of fluctuation and decrease the parameter gradually. Annealing methods face with a difficulty when a phase transition point exists during the protocol. Then, it is important to develop a method to avoid the phase transition by introducing a new type of fluctuation. By taking the Potts model for instance, we review a way to change the phase transition nature. Although the method described in this paper does not succeed to avoid the phase transition, we believe that the concept of the method will be useful for optimization problems.Comment: 27 pages, 3 figures, revised version will appear in proceedings of Kinki University Quantum Computing Series Vo.

Phase Transition in Potts Model with Invisible States

We study phase transition in the ferromagnetic Potts model with invisible states that are added as redundant states by mean-field calculation and Monte Carlo simulation. Invisible states affect the entropy and the free energy, although they do not contribute to the internal energy. The internal energy and the number of degenerated ground states do not change, if invisible states are introduced into the standard Potts model. A second-order phase transition takes place at finite temperature in the standard $q$-state ferromagnetic Potts model on two-dimensional lattice for $q=2,3$, and 4. However, our present model on two-dimensional lattice undergoes a first-order phase transition with spontaneous $q$-fold symmetry breaking ($q=2,3$, and 4) due to entropy effect of invisible states. We believe that our present model is a fundamental model for analysis of a first-order phase transition with spontaneous discrete symmetry breaking.Comment: 8 pages, 4 figure

Relation between dispersion lines and conductance of telescoped armchair double-wall nanotubes analyzed using perturbation formulas and first-principles calculations

The Landauer's formula conductance of the telescoped armchair nanotubes is calculated with the Hamiltonian defined by first-principles calculations (SIESTA code). Herein, partially extracting the inner tube from the outer tube is called 'telescoping'. It shows a rapid oscillation superposed on a slow oscillation as a function of discrete overlap length $(L-1/2)a$ with an integer variable $L$ and the lattice constant $a$. Considering the interlayer Hamiltonian as a perturbation, we obtain the approximate formula of the amplitude of the slow oscillation as $|A|^2/(|A|^2+\varepsilon^2)$ where $A$ is the effective interlayer interaction and $\varepsilon$ is the band split without interlayer interaction. The approximate formula is related to the Thouless number of the dispersion lines.Comment: 9 figure