  We introduce a formulation of optimal transport problem for distributions on
function spaces, where the stochastic map between functional domains can be
partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt
operator mapping a Hilbert space of functions to another. For numerous machine
learning tasks, data can be naturally viewed as samples drawn from spaces of
functions, such as curves and surfaces, in high dimensions. Optimal transport
for functional data analysis provides a useful framework of treatment for such
domains. { Since probability measures in infinite dimensional spaces generally
lack absolute continuity (that is, with respect to non-degenerate Gaussian
measures), the Monge map in the standard optimal transport theory for finite
dimensional spaces may not exist. Our approach to the optimal transport problem
in infinite dimensions is by a suitable regularization technique -- we restrict
the class of transport maps to be a Hilbert-Schmidt space of operators.} To
this end, we develop an efficient algorithm for finding the stochastic
transport map between functional domains and provide theoretical guarantees on
the existence, uniqueness, and consistency of our estimate for the
Hilbert-Schmidt operator. We validate our method on synthetic datasets and
examine the functional properties of the transport map. Experiments on
real-world datasets of robot arm trajectories further demonstrate the
effectiveness of our method on applications in domain adaptation.
