  To address functional-output regression, we introduce projection learning
(PL), a novel dictionary-based approach that learns to predict a function that
is expanded on a dictionary while minimizing an empirical risk based on a
functional loss. PL makes it possible to use non orthogonal dictionaries and
can then be combined with dictionary learning; it is thus much more flexible
than expansion-based approaches relying on vectorial losses. This general
method is instantiated with reproducing kernel Hilbert spaces of vector-valued
functions as kernel-based projection learning (KPL). For the functional square
loss, two closed-form estimators are proposed, one for fully observed output
functions and the other for partially observed ones. Both are backed
theoretically by an excess risk analysis. Then, in the more general setting of
integral losses based on differentiable ground losses, KPL is implemented using
first-order optimization for both fully and partially observed output
functions. Eventually, several robustness aspects of the proposed algorithms
are highlighted on a toy dataset; and a study on two real datasets shows that
they are competitive compared to other nonlinear approaches. Notably, using the
square loss and a learnt dictionary, KPL enjoys a particularily attractive
trade-off between computational cost and performances.
