Sparse Signal Processing
It is well known that under-determined linear systems of equations may admit infinitely many solutions. Within the field of sparse signal processing, this deficiency is overcome by assuming that the unknown vector is sparse, i.e., that most of its coefficients are zero. Because Newton's method fails to identify the correct solution, new algorithms have to be developed that are capable to distinguish the sparse solution from the other solutions.
Such under-determined linear systems of equations arise quite naturally in nonlinear estimation problems when parameters are discretized onto a grid to reduce the computational complexity. Examples of such problems are joint range and azimuth estimation in MIMO radar imaging or channel estimation in future millimeter wave communication systems.
In mathematical terms, the coefficient structure can be expressed as a union of subspaces constraint on the unknowns. In our research, we develop and analyze algorithms that can identify the correct solution in union of subspaces models that are more general than those arising from the zero structure of the vector. With such general models, we can analyze algorithms for estimation problems that are only partially discretized and where existing methods can be used for the non-discretized parameters.