Biblio

Kernel embeddings of distributions have recently gained significant attention in the machine learning community as a data-driven technique for representing probability distributions. Broadly, these techniques enable efficient computation of expectations by representing integral operators as elements in a reproducing kernel Hilbert space. We apply these techniques to the area of stochastic optimal control theory and present a method to compute approximately optimal policies for stochastic systems with arbitrary disturbances. Our approach reduces the optimization problem to a linear program, which can easily be solved via the Lagrangian dual, without resorting to gradient-based optimization algorithms. We focus on discrete- time dynamic programming, and demonstrate our proposed approach on a linear regulation problem, and on a nonlinear target tracking problem. This approach is broadly applicable to a wide variety of optimal control problems, and provides a means of working with stochastic systems in a data-driven setting.

This brief presents a methodology to develop recursive filters in reproducing kernel Hilbert spaces. Unlike previous approaches that exploit the kernel trick on filtered and then mapped samples, we explicitly define the model recursivity in the Hilbert space. For that, we exploit some properties of functional analysis and recursive computation of dot products without the need of preimaging or a training dataset. We illustrate the feasibility of the methodology in the particular case of the γ-filter, which is an infinite impulse response filter with controlled stability and memory depth. Different algorithmic formulations emerge from the signal model. Experiments in chaotic and electroencephalographic time series prediction, complex nonlinear system identification, and adaptive antenna array processing demonstrate the potential of the approach for scenarios where recursivity and nonlinearity have to be readily combined.