  Multivariate time series are ubiquitous objects in signal processing.
Measuring a distance or similarity between two such objects is of prime
interest in a variety of applications, including machine learning, but can be
very difficult as soon as the temporal dynamics and the representation of the
time series, {\em i.e.} the nature of the observed quantities, differ from one
another. In this work, we propose a novel distance accounting both feature
space and temporal variabilities by learning a latent global transformation of
the feature space together with a temporal alignment, cast as a joint
optimization problem. The versatility of our framework allows for several
variants depending on the invariance class at stake. Among other contributions,
we define a differentiable loss for time series and present two algorithms for
the computation of time series barycenters under this new geometry. We
illustrate the interest of our approach on both simulated and real world data
and show the robustness of our approach compared to state-of-the-art methods.
