  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.
