Cartis, Coralia

Tanner, Jared

Thompson, Andrew J.

Engineering and Physical Sciences Research Council (EPSRC)

2014-10-24T13:19:30Z

2014-10-24T13:19:30Z

2013-07-01

Compressed Sensing (CS) is an emerging paradigm in which signals are recovered from undersampled nonadaptive linear measurements taken at a rate proportional to the signal's true information content as opposed to its ambient dimension. The resulting problem consists in finding a sparse solution to an underdetermined system of linear equations. It has now been established, both theoretically and empirically, that certain optimization algorithms are able to solve such problems. Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2007), which is the focus of this thesis, is an established CS recovery algorithm which is known to be effective in practice, both in terms of recovery performance and computational efficiency. However, theoretical analysis of IHT to date suffers from two drawbacks: state-of-the-art worst-case recovery conditions have not yet been quantified in terms of the sparsity/undersampling trade-off, and also there is a need for average-case analysis in order to understand the behaviour of the algorithm in practice. In this thesis, we present a new recovery analysis of IHT, which considers the fixed points of the algorithm. In the context of arbitrary matrices, we derive a condition guaranteeing convergence of IHT to a fixed point, and a condition guaranteeing that all fixed points are 'close' to the underlying signal. If both conditions are satisfied, signal recovery is therefore guaranteed. Next, we analyse these conditions in the case of Gaussian measurement matrices, exploiting the realistic average-case assumption that the underlying signal and measurement matrix are independent. We obtain asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling trade-off for which recovery is guaranteed. By generalizing the notion of xed points, we extend our analysis to the variable stepsize Normalised IHT (NIHT) (Blumensath and Davies, 2010). For both stepsize schemes, comparison with previous results within this framework shows a substantial quantitative improvement. We also extend our analysis to a related algorithm which exploits the assumption that the underlying signal exhibits tree-structured sparsity in a wavelet basis (Baraniuk et al., 2010). We obtain recovery conditions for Gaussian matrices in a simplified proportional-dimensional asymptotic, deriving bounds on the oversampling rate relative to the sparsity for which recovery is guaranteed. Our results, which are the first in the phase transition framework for tree-based CS, show a further significant improvement over results for the standard sparsity model. We also propose a dynamic programming algorithm which is guaranteed to compute an exact tree projection in low-order polynomial time.

http://hdl.handle.net/1842/9603

en

The University of Edinburgh

J. Blanchard, C. Cartis, J. Tanner, and A. Thompson. Phase transitions for greedy sparse approximation algorithms; extended technical report. Technical Report ERGO 09-010, School of Mathematics, University of Edinburgh, 2009.

J. Blanchard, C. Cartis, J. Tanner, and A. Thompson. Phase transitions for greedy sparse approximation algorithms. Applied and Computational Harmonic Analysis, 30(2):188-203, 2011.

J. Blanchard and A. Thompson. On support sizes of restricted isometry constants. Applied and Computational Harmonic Analysis, 29(3):382-390, 2010.

J. Blanchard and A. Thompson. Pushing the rip phase transition in compressed sensing. In Proceedings of the European Signal Processing Conference, 2010.

A. Thompson. Compressive single-pixel imaging. Technical Report ERGO 11-006, School of Mathematics, University of Edinburgh, 2011.

A. Thompson. Compressive single-pixel imaging. In 2nd IMA Conference on Mathematics in Defence, 2011.

compressed sensing

sparse inverse problems

gradient projection

algorithms

Gaussian random matrices

recovery guarantees

Quantitative analysis of algorithms for compressed signal recovery

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy