Solution to Linear Inverse Problem with MMV having Linearly Varying Sparsity Structure

In this paper, an extension to current algorithms dealing with inverse problem is considered. In stead of that invariant sparse profile of the solution vectors is concerned, we mainly focus on the problem with linearly varying sparse structures. Two methods are proposed to solve the linear inverse problem with the unknown linearly varying sparse structure by using MMV. In order to adapt to the linearly varying sparse profile of the solutions, one method, named LMMV (Linearly-MMV), introduces a new parameter and makes use of circular shift matrix to convert the new problem to the one with invariant sparse profile and new iterative algorithm is derived in principle of existing methods. Another method, named WMMV (Wide-MMV), attributes the change of the sparse structure of the solution vectors to the inaccuracy caused by the chosen dictionary and combines several rows of the dictionary together, which is equivalent to find a lower dimensional sparse solution and in turn gives a more robust algorithm. Numerical experiments with random dictionaries and applications to direction-of-arrival (DOA) estimation verify the validation of the proposed two methods and their superiority to some existing methods is illustrated.