  We propose a supervised principal component regression method for relating
functional responses with high dimensional covariates. Unlike the conventional
principal component analysis, the proposed method builds on a newly defined
expected integrated residual sum of squares, which directly makes use of the
association between functional response and predictors. Minimizing the
integrated residual sum of squares gives the supervised principal components,
which is equivalent to solving a sequence of nonconvex generalized Rayleigh
quotient optimization problems and thus is computationally intractable. To
overcome this computational challenge, we reformulate the nonconvex
optimization problems into a simultaneous linear regression, with a sparse
penalty added to deal with high dimensional predictors. Theoretically, we show
that the reformulated regression problem recovers the same supervised principal
subspace under suitable conditions. Statistically, we establish non-asymptotic
error bounds for the proposed estimators. Numerical studies and an application
to the Human Connectome Project lend further support.
