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）（后者也可视为近似K-SVD算法）——收敛的充分条件。我们证明：若给定一个良好初始化的字典，其与生成字典的距离不超过$1/\log(K)$，或具有特殊结构以确保初始化字典中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率收敛至生成字典。该结论同样适用于稀疏系数支撑集呈非均匀分布的数据模型。这些非均匀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建模。