Iterative soft-thresholding converges linearly

In this article, the convergence of the often used iterative softthresholding algorithm for the solution of linear operator equations in infinite dimensional Hilbert spaces is analyzed in detail. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property. This result quantifies the experience that the iterative soft-thresholding converges very slow because it is argued that the constant of the linear rate is very close to one. Moreover it is shown that the constants can be calculated explicitly if some knowledge on the operator is available (i.e. for compact operators). The proof is based on gradient descent arguments for a generalized projected gradient method and utilizes a new technique, called the Bregman-Taylor distance.