Dynamics of Coordinate Ascent Variational Inference: A Case Study in 2D Ising Models

Variational algorithms have gained prominence over the past two decades as a scalable computational environment for Bayesian inference. In this article, we explore tools from the dynamical systems literature to study the convergence of coordinate ascent algorithms for mean field variational inference. Focusing on the Ising model defined on two nodes, we fully characterize the dynamics of the sequential coordinate ascent algorithm and its parallel version. We observe that in the regime where the objective function is convex, both the algorithms are stable and exhibit convergence to the unique fixed point. Our analyses reveal interesting discordances between these two versions of the algorithm in the region when the objective function is non-convex. In fact, the parallel version exhibits a periodic oscillatory behavior which is absent in the sequential version. Drawing intuition from the Markov chain Monte Carlo literature, we empirically show that a parameter expansion of the Ising model, popularly called the Edward–Sokal coupling, leads to an enlargement of the regime of convergence to the global optima.     

Surface phase transitions induced by electron mediated adatom-adatom interaction

We propose that the indirect adatom-adatom interaction mediated by the conduction electrons of a metallic surface is responsible for the sqrt[3]xsqrt[3]<==>3x3 structural phase transitions observed in Sn/Ge (111) and Pb/Ge (111). When the indirect interaction overwhelms the local stress field imposed by the substrate registry, the system suffers a phonon instability, resulting in a structural phase transition in the adlayer. Our theory is capable of explaining all the salient features of the sqrt[3]xsqrt[3]<==>3x3 transitions observed in Sn/Ge (111) and Pb/Ge (111), and is in principle applicable to a wide class of systems whose surfaces are metallic before the transition.     

Frozen low-spin interface in ultrathin Fe films on Cu(111)

In ultrathin film systems, it is a major challenge to understand how a thickness-driven phase transition proceeds along the cross-sectional direction of the films. We use ultrathin Fe films on Cu(111) as a prototype system to demonstrate how to obtain such information using an in situ scanning tunneling microscope and the surface magneto-optical Kerr effect. The magnetization depth profile of a thickness-driven low-spin to high-spin magnetic phase transition is deduced from the experimental data, which leads us to conclude that a low-spin Fe layer at the Fe/Cu interface stays live upon the phase transition. The magnetically live low-spin phase is believed to be induced by a frozen fcc Fe layer that survives a thickness-driven fcc-->bcc structural transition.     

Radial oscillations of neutron stars in strong magnetic fields

The eigen frequencies of radial pulsations of neutron stars are calculated in a strong magnetic field. At low densities we use the magnetic BPS equation of state (EOS) similar to that obtained by Lai and Shapiro while at high densities the EOS obtained from the relativistic nuclear mean field theory is taken and extended to include strong magnetic field. It is found that magnetized neutron stars support higher maximum mass whereas the effect of magnetic field on radial stability for observed neutron star masses is minimal.     

Shape of the quantum diffusion front

We show that quantum diffusion has well-defined front shape. After an initial transient, the wave packet front (tails) is described by a stretched exponential P(x,t) = A(t)exp(-absolute value of [x/w](gamma)), with 1 < gamma < infinity, where w(t) is the spreading width which scales as w(t) approximately t(beta), with 0 < beta < or = 1. The two exponents satisfy the universal relation gamma = 1/(1-beta). We demonstrate these results through numerical work on one-dimensional quasiperiodic systems and the three-dimensional Anderson model of disorder. We provide an analytical derivation of these relations by using the memory function formalism of quantum dynamics. Furthermore, we present an application to experimental results for the quantum kicked rotor.     

Manifestations of broken symmetry: the surface phases of Ca(2-x)SrxRuO4

The surface structural phases of Ca(2-x)SrxRuO4 are investigated using quantitative low energy electron diffraction. The broken symmetry at the surface enhances the structural instability against the RuO6 rotational distortion while diminishing the instability against the RuO6 tilt distortion occurring within the bulk crystal. As a result, suppressed structural and electronic surface phase transition temperatures are observed, including the appearance of an inherent Mott metal-to-insulator transition for x=0.1 and possible modifications of the surface quantum critical point near x(c) approximately 0.5.