Convergence of alternating minimisation algorithms for dictionary learning

In this paper we derive sufficient conditions for the convergence of two popular alternating minimisation algorithms for dictionary learning - the Method of Optimal Directions (MOD) and Online Dictionary Learning (ODL), which can also be thought of as approximative K-SVD. We show that given a well-behaved initialisation that is either within distance at most $1/\log(K)$ to the generating dictionary or has a special structure ensuring that each element of the initialisation only points to one generating element, both algorithms will converge with geometric convergence rate to the generating dictionary. This is done even for data models with non-uniform distributions on the supports of the sparse coefficients. These allow the appearance frequency of the dictionary elements to vary heavily and thus model real data more closely.