Uppsala University
Department of Information Technology

### Licentiate thesis 2014-003

# On Several Sparsity Related Problems and the Randomized Kaczmarz Algorithm

### Liang Dai

9 April 2014

**Abstract:**
This thesis studies several problems related to recovery and estimation. Specifically, these problems are about sparsity and low-rankness, and the randomized Kaczmarz algorithm. This thesis includes four papers referred to as Paper A, Paper B, Paper C, and Paper D.

Paper A considers how to make use of the fact that the solution to an overdetermined system is sparse. This paper presents a three-stage approach to accomplish the task. We show that this strategy, under the assumptions as made in the paper, achieves the oracle property.

In Paper B, a Hankel-matrix completion problem arising in system theory is studied. The use of the nuclear norm heuristic for this basic problem is considered. Theoretical justification for the case of a single real pole is given. Results show that for the case of a single real pole, the nuclear norm heuristic succeeds in the matrix completion task. Numerical simulations indicate that this result does not always carry over to the case of two real poles.

Paper C discusses a screening approach for improving the computational performance of the Basis Pursuit De-Noising problem. The key ingredient for this work is to make use of an efficient ellipsoid update algorithm. The results of the experiments show that the proposed scheme can improve the overall time complexity for solving the problem.

Paper D studies the choice of the probability distribution for implementing the row-projections in the randomized Kaczmarz algorithm. This relates to an open question in the recent literature. The result proves that a probability distribution resulting in a faster convergence of the algorithm can be found by solving a related Semi-Definite Programming optimization problem.

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