  In this paper, we propose new learning algorithms for approximating
high-dimensional functions using tree tensor networks in a least-squares
setting. Given a dimension tree or architecture of the tensor network, we
provide an algorithm that generates a sequence of nested tensor subspaces based
on a generalization of principal component analysis for multivariate functions.
An optimal least-squares method is used for computing projections onto the
generated tensor subspaces, using samples generated from a distribution
depending on the previously generated subspaces. We provide an error bound in
expectation for the obtained approximation. Practical strategies are proposed
for adapting the feature spaces and ranks to achieve a prescribed error. Also,
we propose an algorithm that progressively constructs the dimension tree by
suitable pairings of variables, that allows to further reduce the number of
samples necessary to reach that error. Numerical examples illustrate the
performance of the proposed algorithms and show that stable approximations are
obtained with a number of samples close to the number of free parameters of the
estimated tensor networks.
