Functional Data Analysis (FDA) by matrix completion

Functional data are complex data objects, such as curves and surfaces, that can be seen as realizations of a random function. Covariance operators are at the core of the analysis of such data, since functional PCA  is the canonical dimension reduction technique used to go from infinite to finite dimensions. In  this talk, we consider the problem of nonparametric estimation of a covariance operator, given a sample of discretly observed functional data, for two different setups. In the first setup, we suppose that the observed data arise as the sum of two uncorrelated components, a smooth one representing the global variations of the data and a rough one representing the local variations of the data, and our focus is to recover the covariance operator of the smooth component. In the second setup, we suppose that the discretly observed functional data are fragments, i.e. that the curves are not observed on the whole domain of definition [0,1] but only on a subinterval of length strictly smaller than 1, and we want to recover the covariance operator on the whole unit square. For each setup, we show that the estimation problem translates to a low-rank matrix completion problem, and construct a nonparametric estimator via rank-constrained least squares.  We illustrate our method by simulation and analysis of real data, and provide theory to show the validity of the method, including consistency and rates of convergence.