In this paper we compare non-linear sampling recovery methods for multivariate function classes. In the first part of the paper we propose square root Lasso with a particular choice of the regularization parameter \(\lambda >0\) as a noise blind decoder which efficiently recovers multivariate functions from random samples. In contrast to basis pursuit denoising the algorithm does not require any additional information on the width of the function class in \(L_\infty \) . We then relate the findings to commonly used linear recovery methods and compare the performance in a model situation, namely periodic multivariate functions with bounded mixed derivative in \(L_{q}\) . The main observation is the fact, that square root Lasso asymptotically outperforms Smolyak’s algorithm (sparse grids) in various situations. For \(q=2\) we even see that square root Lasso outperforms any linear method including recently investigated optimal least squares methods.