  This work aims to give non-asymptotic results for estimating the first
principal component of a multivariate random process. We first define the
covariance function and the covariance operator in the multivariate case. We
then define a projection operator. This operator can be seen as a
reconstruction step from the raw data in the functional data analysis context.
Next, we show that the eigenelements can be expressed as the solution to an
optimization problem, and we introduce the LASSO variant of this optimization
problem and the associated plugin estimator. Finally, we assess the estimator's
accuracy. We establish a minimax lower bound on the mean square reconstruction
error of the eigenelement, which proves that the procedure has an optimal
variance in the minimax sense.
