Sparse Approximation using ℓ1-ℓ2 Minimization and Its Application to Stochastic Collocation

We discuss the properties of sparse approximation using $\ell_1-\ell_2$ minimization. We present several theoretical estimates regarding its recoverability for both sparse and nonsparse signals. We then apply the method to sparse orthogonal polynomial approximations for stochastic collocation, with a focus on the use of Legendre polynomials. We study the recoverability of both the standard $\ell_1-\ell_2$ minimization and Chebyshev weighted $\ell_1-\ell_2$ minimization. It is noted that the Chebyshev weighted version is advantageous only at low dimensions, whereas the standard nonweighted version is preferred in high dimensions. Various numerical examples are presented to verify the theoretical findings.