  We present a method to compute the stochastic reachability safety
probabilities for high-dimensional stochastic dynamical systems. Our approach
takes advantage of a nonparametric learning technique known as conditional
distribution embeddings to model the stochastic kernel using a data-driven
approach. By embedding the dynamics and uncertainty within a reproducing kernel
Hilbert space, it becomes possible to compute the safety probabilities for
stochastic reachability problems as simple matrix operations and inner
products. We employ a convergent approximation technique, random Fourier
features, in order to alleviate the increased computational requirements for
high-dimensional systems. This technique avoids the curse of dimensionality,
and enables the computation of safety probabilities for high-dimensional
systems without prior knowledge of the structure of the dynamics or
uncertainty. We validate this approach on a double integrator system, and
demonstrate its capabilities on a million-dimensional, nonlinear, non-Gaussian,
repeated planar quadrotor system.
