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

Quasi-parametric rates for Sparse Multivariate Functional Principal Components Analysis