  Using representations of functional data can be more convenient and
beneficial in subsequent statistical models than direct observations. These
representations, in a lower-dimensional space, extract and compress information
from individual curves. The existing representation learning approaches in
functional data analysis usually use linear mapping in parallel to those from
multivariate analysis, e.g., functional principal component analysis (FPCA).
However, functions, as infinite-dimensional objects, sometimes have nonlinear
structures that cannot be uncovered by linear mapping. Linear methods will be
more overwhelmed given multivariate functional data. For that matter, this
paper proposes a functional nonlinear learning (FunNoL) method to sufficiently
represent multivariate functional data in a lower-dimensional feature space.
Furthermore, we merge a classification model for enriching the ability of
representations in predicting curve labels. Hence, representations from FunNoL
can be used for both curve reconstruction and classification. Additionally, we
have endowed the proposed model with the ability to address the missing
observation problem as well as to further denoise observations. The resulting
representations are robust to observations that are locally disturbed by
uncontrollable random noises. We apply the proposed FunNoL method to several
real data sets and show that FunNoL can achieve better classifications than
FPCA, especially in the multivariate functional data setting. Simulation
studies have shown that FunNoL provides satisfactory curve classification and
reconstruction regardless of data sparsity.
