An optimization based approach for the blind signal separation problem under new sparsity assumptions.

Signal separation (or signal unmixing) is a critical preprocessing step in major data analysis problems arising in engineering and biomedical fields, where pure signals cannot be directly extracted from its individual sources. Typically, multiple sources that emit signals are located inside a system, and the signals recorded on the surface or exterior of the system. The source signals often overlap and get mixed at the receiving points on the exterior. Signal separation is a mechanism to extract pure source signals from the mixture signals, when direct recordings are impractical. However, not all the mixture signals can be successfully unmixed. Traditional theory assumes source signals to be statistically independent for a fruitful unmixing. The statistical assumption is not only rare in typical engineering and biomedical scenarios, but also non-trivial to justify for the signals of finite length. Therefore, many data analysis problems call on new signal separation methods without the statistical independence assumptions. The new methods provide data scientists with alternate platform in the pursuit of understanding the complex real world systems. The focus of this work is to explore signal separation methods based on geometric assumptions, which heavily rely on the sparse nature of the source signals. This class of assumptions are fairly new in the literature, and are in their infant stages when compared to the conventional statistical assumptions. Basically, the geometric assumptions requires certain level of sparsity in the source signals. Moreover, the level of sparsity, in a quantitative sense, can be precisely defined for a signal of finite length. Currently, there are very few sparse approaches, typically, involving series of optimization problems. For big data analytics, the iterative optimization approaches may not be efficient. Thus, our primary goal will involve developing efficient algorithms for the signal separation problems based on the existing sparsity assumptions. The performance of the proposed algorithm will be computationally evaluated with an existing benchmark from the literature. Furthermore, identifying new conditions that will guarantee perfect separation will be an additional goal of this work. The new conditions will enlarge the scope and applicability of the sparsity based methods.