Entropic herding

Herding is a deterministic algorithm used to generate data points regarded as random samples satisfying input moment conditions. This algorithm is based on a high-dimensional dynamical system and rooted in the maximum entropy principle of statistical inference. We propose an extension, entropic herding, which generates a sequence of distributions instead of points. We derived entropic herding from an optimization problem obtained using the maximum entropy principle. Using the proposed entropic herding algorithm as a framework, we discussed a closer connection between the herding and maximum entropy principle. Specifically, we interpreted the original herding algorithm as a tractable version of the entropic herding, the ideal output distribution of which is mathematically represented. We further discussed how the complex behavior of the herding algorithm contributes to optimization. We argued that the proposed entropic herding algorithm extends the herding to probabilistic modeling. In contrast to the original herding, the entropic herding can generate a smooth distribution such that both efficient probability density calculation and sample generation become possible. To demonstrate the viability of these arguments in this study, numerical experiments were conducted, including a comparison with other conventional methods, on both synthetic and real data.