  Our research proposes a novel method for reducing the dimensionality of
functional data, specifically for the case where the response is a scalar and
the predictor is a random function. Our method utilizes distance covariance,
and has several advantages over existing methods. Unlike current techniques
which require restrictive assumptions such as linear conditional mean and
constant covariance, our method has mild requirements on the predictor.
Additionally, our method does not involve the use of the unbounded inverse of
the covariance operator. The link function between the response and predictor
can be arbitrary, and our proposed method maintains the advantage of being
model-free, without the need to estimate the link function. Furthermore, our
method is naturally suited for sparse longitudinal data. We utilize functional
principal component analysis with truncation as a regularization mechanism in
the development of our method. We provide justification for the validity of our
proposed method, and establish statistical consistency of the estimator under
certain regularization conditions. To demonstrate the effectiveness of our
proposed method, we conduct simulation studies and real data analysis. The
results show improved performance compared to existing methods.
