  In this paper, we propose a low-rank approximation method based on discrete
least-squares for the approximation of a multivariate function from random,
noisy-free observations. Sparsity inducing regularization techniques are used
within classical algorithms for low-rank approximation in order to exploit the
possible sparsity of low-rank approximations. Sparse low-rank approximations
are constructed with a robust updated greedy algorithm which includes an
optimal selection of regularization parameters and approximation ranks using
cross validation techniques. Numerical examples demonstrate the capability of
approximating functions of many variables even when very few function
evaluations are available, thus proving the interest of the proposed algorithm
for the propagation of uncertainties through complex computational models.
