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Sparse Representation and its Applications in Multi-Sensor Problems

Malek-Mohammadi, Mohammad Reza | 2015

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 50523 (05)
  4. University: Sharif University of Technology
  5. Department: Electrical Engineering
  6. Advisor(s): Babaie-Zade, Massoud
  7. Abstract:
  8. Recovery of low-rank matrices from compressed linear measurements is an extension for the more well-known topic of recovery of sprse vectors from underdetermined measurements.Since the natural approach (i.e., rank minimization) for recovery of low-rank matrices is generally NP-hard, several alternatives have been proposed. However, there is a large gap between what can be achieved from these alternatives and the natural approach in terms of maximum rank of the unique solutions and the error of recovery. To narrow this gap, two novel algorithms are proposed. The main idea of both algorithms is to closely approximate the rank with a smooth function of singular values and then minimize the consequent approximation subject to the linear constraints. The fundamental diffrerence between these two algorithms, however, is in the class of functions used to approximate the rank function.As the rank approximating functions are nonconvex, to decrease the chance of getting trapped in local solutions, a series of optimization programs is solved. Initially, a rough approximation of the rank function subject to the affine constraints is optimized. As the algorithm proceeds, finer approximations of the rank are optimized and the solver is initialized with the solution of the previous approximation until reaching the desired accuracy.We theoreticall show that the sequence of the solutions of the approximating functions converges to the minimum rank solution. For the first algorithm which is called SRF, we use a class of functions which can be easily optimized. For the second algorithm which is named ICRA, a class of subadditive functions is used that leads to higher performance at the cost of increased computational complexity. Exploiting numerical simulations, we show that in recovering low-rank matrices from compressed measurements and in completing partially observed matrices, the proposed algorithms considerably and consistently outperform some of the state-of-the-art algorithms.Next, the idea of ICRA is translated to sparse vector recovery. The new algorithm which is called SCSA is based on a successively accuracy-increasing approximation of the ℓ0 norm.The approximations are realized with a certain class of concave functions that aggressively induce sparsity and their closeness to the ℓ0 norm can be controlled. We prove that the series of the approximations asymptotically coincides with the ℓ1 and ℓ0 norms when the approximation accuracy changes from the worst fitting to the best fitting. To have a significant contribution with respect to ICRA, we extend SCSA to the noisy setting and by exploiting the iterative thresholding technique, we keep its computational complexity at a reasonable level. For a particular function in the class of the approximating functions, we derive the closed-form thresholding operator which is characterized with the Lambert W function. Our extensive numerical simulations indicate that the proposed algorithm closely follows the performance of the oracle estimator for a range of sparsity levels wider than those of the state-of-the-art algorithms.The next part of this thesis focuses on presenting a few solutions to a few problems in mulisensor based application. Particularly, using sparse vector and low-rank matrix recovery algorithms, we propose a novel method which can estimate the emitting directions of possibly correlated or even coherent sources in an array of antennas where the noise is spatially correlated. We also apply the same idea to detect the presence of primary users in similar environments
  9. Keywords:
  10. Matrix Completion ; Compressive Sensing ; Sparse Recovery ; Low-Rank Matrix ; Multisensory Systems ; Affine Rank Minimization

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