  The joint alignment of multivariate functional data plays an important role
in various fields such as signal processing, neuroscience and medicine,
including the statistical analysis of data from wearable devices. Traditional
methods often ignore the phase variability and instead focus on the variability
in the observed amplitude. We present a novel method for joint alignment of
multivariate quasi-periodic functions using deep neural networks, decomposing,
but retaining all the information in the data by preserving both phase and
amplitude variability. Our proposed neural network uses a special activation of
the output that builds on the unit simplex transformation, and we utilize a
loss function based on the Fisher-Rao metric to train our model. Furthermore,
our method is unsupervised and can provide an optimal common template function
as well as subject-specific templates. We demonstrate our method on two
simulated datasets and one real example, comprising data from 12-lead 10s
electrocardiogram recordings.
