  This work studies the multi-task functional linear regression models where
both the covariates and the unknown regression coefficients (called slope
functions) are curves. For slope function estimation, we employ penalized
splines to balance bias, variance, and computational complexity. The power of
multi-task learning is brought in by imposing additional structures over the
slope functions. We propose a general model with double regularization over the
spline coefficient matrix: i) a matrix manifold constraint, and ii) a composite
penalty as a summation of quadratic terms. Many multi-task learning approaches
can be treated as special cases of this proposed model, such as a reduced-rank
model and a graph Laplacian regularized model. We show the composite penalty
induces a specific norm, which helps to quantify the manifold curvature and
determine the corresponding proper subset in the manifold tangent space. The
complexity of tangent space subset is then bridged to the complexity of
geodesic neighbor via generic chaining. A unified convergence upper bound is
obtained and specifically applied to the reduced-rank model and the graph
Laplacian regularized model. The phase transition behaviors for the estimators
are examined as we vary the configurations of model parameters.
