Incorporating machine learning techniques into optimization problems and solvers attracts increasing attention. Given a particular type of optimization problem that needs to be solved repeatedly, machine learning techniques can find some features for this category of optimization and develop algorithms with excellent performance. This thesis deals with algorithms and convergence analysis in learning-based optimization in three aspects: learning dictionaries, learning optimization solvers and learning regularizers.
Learning dictionaries for sparse coding is significant for signal processing. Convolutional sparse coding is a form of sparse coding with a structured, translation invariant dictionary. Most convolutional dictionary learning algorithms to date operate in the batch mode, requiring simultaneous access to all training images during the learning process, which results in very high memory usage, and severely limits the training data size that can be used. I proposed two online convolutional dictionary learning algorithms that offered far better scaling of memory and computational cost than batch methods and provided a rigorous theoretical analysis of these methods.
Learning fast solvers for optimization is a rising research topic. In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theoretical understanding is still immature, which prevents us from fully utilizing the power of neural networks. I studied unfolded ISTA (Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery and established its convergence. Based on the properties of parameters required by convergence, the model can be significantly simplified and, consequently, has much less training cost and better recovery performance.
Learning regularizers or priors improves the performance of optimization solvers, especially for signal and image processing tasks.
Plug-and-play (PnP) is a non-convex framework that integrates modern priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this thesis, the theoretical convergence of PnP-FBS and PnP-ADMM was established, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. Furthermore, real spectral normalization was proposed for training deep learning-based denoisers to satisfy the proposed Lipschitz condition.
