  This paper gives a comprehensive treatment of the convergence rates of
penalized spline estimators for simultaneously estimating several leading
principal component functions, when the functional data is sparsely observed.
The penalized spline estimators are defined as the solution of a penalized
empirical risk minimization problem, where the loss function belongs to a
general class of loss functions motivated by the matrix Bregman divergence, and
the penalty term is the integrated squared derivative. The theory reveals that
the asymptotic behavior of penalized spline estimators depends on the
interesting interplay between several factors, i.e., the smoothness of the
unknown functions, the spline degree, the spline knot number, the penalty
order, and the penalty parameter. The theory also classifies the asymptotic
behavior into seven scenarios and characterizes whether and how the minimax
optimal rates of convergence are achievable in each scenario.
