Quantitative Recovery Conditions for Tree-Based Compressed Sensing

Abstract

As shown by Blumensath and Davies (2009) and 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. In the context of Gaussian matrices, we apply this framework to existing worst-case analysis of the iterative tree projection (ITP) algorithm, which makes use of the tree-based restricted isometry property (RIP). Within the same framework, we then obtain quantitative results based on a new method of analysis, which considers the fixed points of the algorithm. By exploiting the realistic average-case assumption that the measurements are statistically independent of the signal, we obtain significant quantitative improvements when compared with the tree-based RIP analysis. 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. For example, we prove that exact recovery of binary tree-based signals from noiseless Gaussian measurements is asymptotically guaranteed for ITP with constant stepsize provided $n\geq 50k$ . All our results extend to the more realistic case in which measurements are corrupted by noise.