The Roles of Sparseness and Robust Linear Algebra in Simultaneous-Source Separation

Simultaneous-source separation algorithms typically address the underdetermined nature of the separation problem by assuming that the separated data have a sparse representation over some dictionary of basis atoms. Typically, these basis atoms are linear and are localised in the data domain. When the data exhibit different curvature to the basis atoms, this assumption breaks down and enforcing sparseness generally leads to some degree of separation error.

Extending or modifying the dictionary to provide a more efficacious representation of the data is both difficult and expensive given the complexity of real data. Instead, we consider a combination of robust linear algebra and a relaxation of the sparseness conditions to provide an appropriate representation of the data. In the presence of curvature, the desired contributions to the robust correlations vary from shot to shot. We compare standard correlations with robust correlations, both ignoring and accounting for this variation when suppressing interference. A simple synthetic example shows that, whilst either form of robust correlation is vastly superior to the standard form, the robust form that accounts for the variability of the contributions provides a significant further improvement (for curved events) in the estimated correlations and the corresponding estimate of the separated data.