  Time series alignment methods call for highly expressive, differentiable and
invertible warping functions which preserve temporal topology, i.e
diffeomorphisms. Diffeomorphic warping functions can be generated from the
integration of velocity fields governed by an ordinary differential equation
(ODE). Gradient-based optimization frameworks containing diffeomorphic
transformations require to calculate derivatives to the differential equation's
solution with respect to the model parameters, i.e. sensitivity analysis.
Unfortunately, deep learning frameworks typically lack
automatic-differentiation-compatible sensitivity analysis methods; and implicit
functions, such as the solution of ODE, require particular care. Current
solutions appeal to adjoint sensitivity methods, ad-hoc numerical solvers or
ResNet's Eulerian discretization. In this work, we present a closed-form
expression for the ODE solution and its gradient under continuous
piecewise-affine (CPA) velocity functions. We present a highly optimized
implementation of the results on CPU and GPU. Furthermore, we conduct extensive
experiments on several datasets to validate the generalization ability of our
model to unseen data for time-series joint alignment. Results show significant
improvements both in terms of efficiency and accuracy.
