  Non-parametric inference for functional data over two-dimensional domains
entails additional computational and statistical challenges, compared to the
one-dimensional case. Separability of the covariance is commonly assumed to
address these issues in the densely observed regime. Instead, we consider the
sparse regime, where the latent surfaces are observed only at few irregular
locations with additive measurement error, and propose an estimator of
covariance based on local linear smoothers. Consequently, the assumption of
separability reduces the intrinsically four-dimensional smoothing problem into
several two-dimensional smoothers and allows the proposed estimator to retain
the classical minimax-optimal convergence rate for two-dimensional smoothers.
Even when separability fails to hold, imposing it can be still advantageous as
a form of regularization. A simulation study reveals a favorable bias-variance
trade-off and massive speed-ups achieved by our approach. Finally, the proposed
methodology is used for qualitative analysis of implied volatility surfaces
corresponding to call options, and for prediction of the latent surfaces based
on information from the entire data set, allowing for uncertainty
quantification. Our cross-validated out-of-sample quantitative results show
that the proposed methodology outperforms the common approach of pre-smoothing
every implied volatility surface separately.
