  This paper is concerned with the approximation of high-dimensional functions
in a statistical learning setting, by empirical risk minimization over model
classes of functions in tree-based tensor format. These are particular classes
of rank-structured functions that can be seen as deep neural networks with a
sparse architecture related to the tree and multilinear activation functions.
For learning in a given model class, we exploit the fact that tree-based tensor
formats are multilinear models and recast the problem of risk minimization over
a nonlinear set into a succession of learning problems with linear models.
Suitable changes of representation yield numerically stable learning problems
and allow to exploit sparsity. For high-dimensional problems or when only a
small data set is available, the selection of a good model class is a critical
issue. For a given tree, the selection of the tuple of tree-based ranks that
minimize the risk is a combinatorial problem. Here, we propose a rank
adaptation strategy which provides in practice a good convergence of the risk
as a function of the model class complexity. Finding a good tree is also a
combinatorial problem, which can be related to the choice of a particular
sparse architecture for deep neural networks. Here, we propose a stochastic
algorithm for minimizing the complexity of the representation of a given
function over a class of trees with a given arity, allowing changes in the
topology of the tree. This tree optimization algorithm is then included in a
learning scheme that successively adapts the tree and the corresponding
tree-based ranks. Contrary to classical learning algorithms for nonlinear model
classes, the proposed algorithms are numerically stable, reliable, and require
only a low level expertise of the user.
