Quasi-parametric rates for Sparse Multivariate Functional Principal Components Analysis

TL;DR

This paper introduces a quasi-parametric approach for estimating the first principal component in multivariate functional data, providing non-asymptotic guarantees and an optimal LASSO-based estimator.

Contribution

It develops a novel LASSO-based optimization framework for multivariate FPCA and establishes minimax optimality of the estimator's variance.

Findings

The estimator achieves minimax optimal mean square error.

The LASSO variant effectively reconstructs the principal component.

Non-asymptotic bounds are provided for the estimation accuracy.

Abstract

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.