Parameter Estimation Using Sparse Modeling: Algorithms and Performance Analysis
Licentiate thesis, 2012
The idea of representing a signal in a classical computing machine has played a central role in the field of signal processing. The last two decades have witnessed an important breakthrough in this by taking all possible linear transforms and domains into account. The current observations show the possibility of reconstructing a sparse signal by few measurements through linear transforms without the knowledge of the subspace where the signal resides.
This work is devoted to the application of such compressive sensing techniques to estimate a set of parameters. We try to address the main conventional ideas of estimation, especially as a regression problem, and connect these ideas to the recently developed technique by domain sparsity. We also review the conventional method of applying the so called Least Absolute Shrinkage and Selection Operator (LASSO) technique to solve estimation by domain sparsity, which looks inappropriate as a continuous estimation solution. In return, we try to develop a framework for the continuous estimation and address its unsolved problems to a concerned reader.
We also introduce a practical method of implementing the continuous LASSO as a successful attempt to solve convex variational problems. We introduce this method in the context of Direction of Arrival (DOA) estimation using an array of sensors by spatial sparsity, which gives us the possibility of analyzing the aforementioned Compressive Sensing (CS) techniques from a different perspective of statistics. The introductory parts contain the essential issues in DOA estimation, which are more or less common in all regression problems. We also review the Bayesian aspects of the LASSO based estimation briefly.
convex variational optimization