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.

翻译：本文推导了字典学习中两种流行的交替最小化算法——最优方向法（MOD）和在线字典学习（ODL，亦可视为近似K-SVD）——收敛的充分条件。我们证明：若初始值具有良好的性质，即要么与生成字典的距离不超过$1/\log(K)$，要么具有特殊结构确保初始值的每个元素仅指向一个生成元素，则两种算法均能以几何收敛速度收敛至生成字典。该结论甚至适用于稀疏系数支持集上非均匀分布的数据模型。这类分布允许字典元素出现频率存在显著差异，从而更贴近真实数据的特性。