Sparsity and Independence: Balancing Two Objectives in Optimization for Source Separation with Application to fMRI Analysis

Because of its wide applicability in various disciplines, blind source separation (BSS), has been an active area of research. For a given dataset, BSS provides useful decompositions under minimum assumptions typically by making use of statistical properties—types of diversity—of the data. Two popular types of diversity that have proven useful for many applications are statistical independence and sparsity. Although many methods have been proposed for the solution of the BSS problem that take either the statistical independence or the sparsity of the data into account, there is no unified method that can take into account both types of diversity simultaneously. In this work, we provide a mathematical framework that enables direct control over the influence of these two types of diversity and apply the proposed framework to the development of an effective ICA algorithm that can jointly exploit independence and sparsity. In addition, due to its importance in biomedical applications, we propose a new model reproducibility framework for the evaluation of the proposed algorithm. Using simulated functional magnetic resonance imaging (fMRI) data, we study the trade-offs between the use of sparsity versus independence in terms of the separation accuracy and reproducibility of the algorithm and provide guidance on how to balance these two objectives in real world applications where the ground truth is not available.