  We propose an extension of a special form of gradient descent -- in the
literature known as linearised Bregman iteration -- to a larger class of
non-convex functions. We replace the classical (squared) two norm metric in the
gradient descent setting with a generalised Bregman distance, based on a
proper, convex and lower semi-continuous function. The algorithm's global
convergence is proven for functions that satisfy the Kurdyka-\L ojasiewicz
property. Examples illustrate that features of different scale are being
introduced throughout the iteration, transitioning from coarse to fine. This
coarse-to-fine approach with respect to scale allows to recover solutions of
non-convex optimisation problems that are superior to those obtained with
conventional gradient descent, or even projected and proximal gradient descent.
The effectiveness of the linearised Bregman iteration in combination with early
stopping is illustrated for the applications of parallel magnetic resonance
imaging, blind deconvolution as well as image classification with neural
networks.
