  We introduce a novel algorithm for estimating optimal parameters of
linearized assignment flows for image labeling. An exact formula is derived for
the parameter gradient of any loss function that is constrained by the linear
system of ODEs determining the linearized assignment flow. We show how to
efficiently evaluate this formula using a Krylov subspace and a low-rank
approximation. This enables us to perform parameter learning by Riemannian
gradient descent in the parameter space, without the need to backpropagate
errors or to solve an adjoint equation. Experiments demonstrate that our method
performs as good as highly-tuned machine learning software using automatic
differentiation. Unlike methods employing automatic differentiation, our
approach yields a low-dimensional representation of internal parameters and
their dynamics which helps to understand how assignment flows and more
generally neural networks work and perform.
