  This paper provides answers to an open problem: given a nonlinear data-driven
dynamical system model, e.g., kernel conditional mean embedding (CME) and
Koopman operator, how can one propagate the ambiguity sets forward for multiple
steps? This problem is the key to solving distributionally robust control and
learning-based control of such learned system models under a data-distribution
shift. Different from previous works that use either static ambiguity sets,
e.g., fixed Wasserstein balls, or dynamic ambiguity sets under known piece-wise
linear (or affine) dynamics, we propose an algorithm that exactly propagates
ambiguity sets through nonlinear data-driven models using the Koopman operator
and CME, via the kernel maximum mean discrepancy geometry. Through both
theoretical and numerical analysis, we show that our kernel ambiguity sets are
the natural geometric structure for the learned data-driven dynamical system
models.
