The Mathematical Institute, University of Oxford, Eprints Archive

Quantitative recovery conditions for tree-based compressed sensing

Cartis, Coralia and Thompson, Andrew (2015) Quantitative recovery conditions for tree-based compressed sensing. Technical Report. Unspecified. (Submitted)

[img]
Preview
PDF
501kB

Abstract

As shown in (Blumensath & Davies (2009), Baraniuk et al. (2010)), signals whose wavelet coefficients exhibit a rooted tree structure can be recovered -- using specially-adapted compressed sensing algorithms -- from just $n=\mathcal{O}(k)$ measurements, where $k$ is the sparsity of the signal. Motivated by these results, we introduce a simplified proportional-dimensional asymptotic framework which enables the quantitative evaluation of recovery guarantees for tree-based compressed sensing algorithms. We consider the Iterative Tree Projection (ITP) algorithm (Blumensath & Davies (2009), Baraniuk et al. (2010)), with a constant and a variable/practically-efficient stepsize scheme, respectively. In the context of Gaussian matrices, we apply our simplified asymptotic framework to existing worst-case analysis of ITP, which makes use of the tree-based Restricted Isometry Property (RIP). Our results have a refreshingly simple interpretation, explicitly determining a bound on the number of measurements that are required as a multiple of the sparsity. In particular, we prove that exact recovery of binary tree-based signals from noiseless Gaussian measurements is asymptotically guaranteed for ITP provided $n\geq 115k$ (constant stepsize) and $n\geq 683k$ (variable stepsize).

Within the same framework, we then obtain quantitative results based on a new method of analysis, recently introduced in (Cartis & Thompson, IEEE Information Theory, 2015), which considers the fixed points of the same ITP algorithmic variants. By exploiting the realistic average-case assumption that the measurements are statistically independent of the signal, we obtain significant quantitative improvements when compared to the tree-based RIP analysis; in this case, exact recovery of binary tree-based signals from noiseless Gaussian measurements is asymptotically guaranteed for ITP provided $n\geq 50k$ (constant stepsize) and $n\geq 55k$ (variable stepsize). All our results are also extended to the more realistic case in which measurements are corrupted by noise.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Numerical Analysis Group
ID Code:1911
Deposited By: Helen Taylor
Deposited On:24 Oct 2015 08:34
Last Modified:24 Oct 2015 08:34

Repository Staff Only: item control page