2016/12/08 by Martin Benning, Benning, Martin, Marta M. Betcke +5
Mathematics · Physics and Astronomy · #49M37 #65K05 #65K10 #90C26 #90C30 #FOS: Mathematics #G.1.0 #G.1.6 #Geometric Analysis and Curvature Flows #Numerical methods in inverse problems #Optimization and Control (math.OC) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.1612.02506
openalex publication_date 2016/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss a special form of gradient descent that in the literature has become known as the so-called linearised Bregman iteration. The idea is to replace the classical (squared) two norm metric in the gradient descent setting with a generalised Bregman distance, based on a more general proper, convex and lower semi-continuous functional. Gradient descent as well as the entropic mirror descent by Nemirovsky and Yudin are special cases, as is a specific form of non-linear Landweber iteration introduced by Bachmayr and Burger. We are going to analyse the linearised Bregman iteration in a setting where the functional we want to minimise is neither necessarily Lipschitz-continuous (in the classical sense) nor necessarily convex, and establish a global convergence result under the additional assumption that the functional we wish to minimise satisfies the so-called Kurdyka-Łojasiewicz property.