A least-squares method for sparse low rank approximation of multivariate functions
This work addresses the challenge of uncertainty propagation in complex computational models, offering a domain-specific solution that is incremental in nature.
The paper tackles the problem of approximating multivariate functions from limited, noise-free observations by proposing a sparse low-rank approximation method using discrete least-squares and sparsity-inducing regularization. The result is a robust greedy algorithm that effectively handles high-dimensional functions with very few evaluations, as demonstrated in numerical examples.
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.