Simple Alternating Minimization Provably Solves Complete Dictionary Learning (2210.12816v1)

Abstract: This paper focuses on complete dictionary learning problem, where the goal is to reparametrize a set of given signals as linear combinations of atoms from a learned dictionary. There are two main challenges faced by theoretical and practical studies of dictionary learning: the lack of theoretical guarantees for practically-used heuristic algorithms, and their poor scalability when dealing with huge-scale datasets. Towards addressing these issues, we show that when the dictionary to be learned is orthogonal, that an alternating minimization method directly applied to the nonconvex and discrete formulation of the problem exactly recovers the ground truth. For the huge-scale, potentially online setting, we propose a minibatch version of our algorithm, which can provably learn a complete dictionary from a huge-scale dataset with minimal sample complexity, linear sparsity level, and linear convergence rate, thereby negating the need for any convex relaxation for the problem. Our numerical experiments showcase the superiority of our method compared with the existing techniques when applied to tasks on real data.