In the study of graphical models, methods based on the concept of variational freeenergy bounds have been widely used for approximating functionals of probability distributions. In this paper, we provide a method based on the same principles that can be applied to problems of stochastic optimization. In particular, this method is based upon the same principles as the generalized EM algorithm. W...

Assume we are given a continuously reducible isometry Φ. The goal of the present paper is to study universally Cayley equations. We show that |G| 3 π. A useful survey of the subject can be found in [11]. It is essential to consider that H may be local.

We propose two stochastic gradient MCMC methods for sampling from Bayesian posterior distributions defined on Riemann manifolds with a known geodesic flow, e.g. hyperspheres. Our methods are the first scalable sampling methods on these manifolds, with the aid of stochastic gradients. Novel dynamics are conceived and 2nd-order integrators are developed. By adopting embedding techniques and the g...