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Greedy algorithms for reconstruction of high-dimensional sparse vectors

Siaminou Ioanna

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Summary

Reconstruction of signals from measured data is often encountered in various fields of science. However, the dimension of the target signal is often much larger than the number of the collected measurements. In these cases, signal reconstruction is practically impossible in general. Luckily, by assuming that the signal we wish to reconstruct has certain structure, the reconstruction becomes feasible. In Compressed Sensing, we deal with the system y = Ax, where the so-called measurement matrix A has dimensions (m x n), with m < n. In this area, the notion of sparsity is used as a constraint on the target signal x. In this thesis, we concentrate on greedy algorithms, studied extensively in the literature, and the conditions that guarantee successful reconstruction. First, we provide a theoretical background of Compressed Sensing and, afterwards, we proceed with the presentation and analysis of greedy algorithms, such as Orthogonal Matching Pursuit (OMP) and Compressive Sampling Matching Pursuit (CoSaMP). We complement our presentation with numerical experiments, using as performance metric the relative signal reconstruction error. Then, we investigate the extension of sparse vector reconstruction in non-linear scenarios. For this purpose, we consider a greedy algorithm, the Gradient Support Pursuit (GraSP), which is an extension of CoSaMP. We present the conditions that must be satisfied in this framework for successful reconstruction, and compare the performance of GraSP to LASSO, of the GLMnet package, for the logistic model.Finally, we propose a method for non-linear scenarios inspired by GraSP and OMP, test it for the logistic model, and compare the results to those of GraSP and GLMnet.

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