Analysis of Fast Structured Dictionary Learning
This work provides incremental theoretical insights for signal processing and imaging applications using sparsity-based models.
The paper tackles the convergence analysis of an alternating minimization algorithm for structured unitary sparsifying operator learning, establishing local linear convergence to the underlying data model under mild assumptions, with numerical simulations confirming robustness to initialization.
Sparsity-based models and techniques have been exploited in many signal processing and imaging applications. Data-driven methods based on dictionary and sparsifying transform learning enable learning rich image features from data, and can outperform analytical models. In particular, alternating optimization algorithms have been popular for learning such models. In this work, we focus on alternating minimization for a specific structured unitary sparsifying operator learning problem, and provide a convergence analysis. While the algorithm converges to the critical points of the problem generally, our analysis establishes under mild assumptions, the local linear convergence of the algorithm to the underlying sparsifying model of the data. Analysis and numerical simulations show that our assumptions hold for standard probabilistic data models. In practice, the algorithm is robust to initialization.