Optimal decision making under uncertainty is critical for efficient control and optimization
of complex systems. However, many current techniques encounter the curse-of-dimensionality
for systems with too many state variables. In this talk, we will present a framework based on the
emerging area of tensor decompositions that mitigates this curse-of-dimensionality. First we de-
scribe the tensor-train decomposition, which provides efficient compression of high dimensional
arrays. Next, we describe our recent work utilizing the tensor-train decomposition for optimal
stochastic control, and we simulate the resulting optimal feedback control on an underactuated
motion planning problem. In the second part of the talk, we will describe the construction of
our newly developed continuous extension to the tensor-train decomposition. The resulting ap-
proximation, termed the function-train, is obtained using an algorithm designed in the spirit of
continuous computation. It involves computing “continuous” matrix-factorizations such as the
LU and QR decomposition of univariate functions. Finally, we will show how this new approx-
imation can be used to integrate high dimensional and discontinuous functions in polynomial
time with dimension . We will also show its approximation properties on a parameteric elliptic
PDE.