Functional Autoencoder for Smoothing and Representation Learning
Sidi Wu Cédric Beaulac Jiguo Cao
A common pipeline in functional data analysis is to first convert the
discretely observed data to smooth functions, and then represent the functions
by a finite-dimensional vector of coefficients summarizing the information.
Existing methods for data smoothing and dimensional reduction mainly focus on
learning the linear mappings from the data space to the representation space,
however, learning only the linear representations may not be sufficient. In
this study, we propose to learn the nonlinear representations of functional
data using neural network autoencoders designed to process data in the form it
is usually collected without the need of preprocessing. We design the encoder
to employ a projection layer computing the weighted inner product of the
functional data and functional weights over the observed timestamp, and the
decoder to apply a recovery layer that maps the finite-dimensional vector
extracted from the functional data back to functional space using a set of
predetermined basis functions. The developed architecture can accommodate both
regularly and irregularly spaced data. Our experiments demonstrate that the
proposed method outperforms functional principal component analysis in terms of
prediction and classification, and maintains superior smoothing ability and
better computational efficiency in comparison to the conventional autoencoders
under both linear and nonlinear settings.