Iterative Methods Based on Soft Thresholding of Hierarchical Tensors

We construct a soft thresholding operation for rank reduction in 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 adjusts the thresholding parameters, by an a posteriori criterion requiring only 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 effectiveness of the scheme is demonstrated in numerical experiments.