  We study an optimization problem related to the approximation of given data
by a linear combination of transformed modes. In the simplest case, the
optimization problem reduces to a minimization problem well-studied in the
context of proper orthogonal decomposition. Allowing transformed modes in the
approximation renders this approach particularly useful to compress data with
transported quantities, which are prevalent in many flow applications. We prove
the existence of a solution to the infinite-dimensional optimization problem.
Towards a numerical implementation, we compute the gradient of the cost
functional and derive a suitable discretization in time and space. We
demonstrate the theoretical findings with three challenging numerical examples.
