Explicit versus implicit source estimation for blind multiple input single output system identification

Sparsely-activated time series are found in many physical systems. In these cases, the signals can be approximated by convolution of sparse sources with a set of shift-invariant filters. When there is access to only one sensor, such that there is a single observation signal, identifying the source signals appears to be an ill-posed problem, but for very sparse sources it is still possible to learn the system. We discuss analysis techniques for sparsely activated signals, which retrieve sparse sources given the filters, and identify conditions when algorithms based on independent component analysis (ICA) and sparse coding can blindly estimate filters from a single noisy time-series. Many qualitative results have been made for learning shift-invariant bases on natural signals, but for a thorough understanding of the effect of sparsity, we quantitatively analyze results on synthetic examples, comparing how ICA and shift-invariant sparse coding approaches perform for multiple-source blind system identification.