Iterative Methods Based on Soft Thresholding of Hierarchical Tensors

We construct a soft thresholding operation for rank reduction of hierarchical tensors and subsequently consider its use in iterative thresholding methods, in particular for the solution of discretized high-dimensional elliptic problems. The proposed method for the latter case automatically adjusts the thresholding parameters, based only on bounds on the spectrum of the operator, such that the arising tensor ranks of the resulting iterates remain quasi-optimal with respect to the algebraic or exponential-type decay of the hierarchical singular values of the true solution. In addition, we give a modified algorithm using inexactly evaluated residuals that retains these features. The practical efficiency of the scheme is demonstrated in numerical experiments.