AbstractsEngineering

This thesis introduces the usage of non-convex based regularizers to solve the underdetermined MEG inverse problem.The signal to be reconstructed is considered to have a structure which entails group-wise sparsity and within group sparsity among its covariates. We discuss the usage of ¿2 norm regularization and smoothed ¿0 (SL0) norm regularization to impose group-wise and within group sparsity respectively. In addition, we introduce a novel criterion which if satisfied, guarantees global optimality while solving this non-convex optimization problem. We use proximal gradient descent as the method of optimization as it promises faster convergence rates. Initially, we show that our algorithm successfully recovers sparse signals with a smaller number of measurements than the conventional ¿1 regularization framework. We also support this claim using MEG source localization simulations and extend the reconstruction for both stationary and non-stationary signals. Next, we formulate a global convergence analysis for the novel algorithm. Finally, we incorporate novel information criteria techniques and concepts of duality to find the best set of regularization parameters and a proper stopping criterion respectively. We were able to successfully illustrate that the regularization parameters (models) with lower information criteria performs better than the ones with higher information criteria. Also, concepts of duality provides the necessary tools to determine when to stop the algorithm, which is an important contribution considering the non- differentiability of the objective function. Advisors/Committee Members: Fan, H. Howard (Committee Chair).