  Fully nonparametric methods for regression from functional data have poor
accuracy from a statistical viewpoint, reflecting the fact that their
convergence rates are slower than nonparametric rates for the estimation of
high-dimensional functions. This difficulty has led to an emphasis on the
so-called functional linear model, which is much more flexible than common
linear models in finite dimension, but nevertheless imposes structural
constraints on the relationship between predictors and responses. Recent
advances have extended the linear approach by using it in conjunction with link
functions, and by considering multiple indices, but the flexibility of this
technique is still limited. For example, the link may be modeled parametrically
or on a grid only, or may be constrained by an assumption such as monotonicity;
multiple indices have been modeled by making finite-dimensional assumptions. In
this paper we introduce a new technique for estimating the link function
nonparametrically, and we suggest an approach to multi-index modeling using
adaptively defined linear projections of functional data. We show that our
methods enable prediction with polynomial convergence rates. The finite sample
performance of our methods is studied in simulations, and is illustrated by an
application to a functional regression problem.
