# Quantitative recovery conditions for tree-based compressed sensing

As shown in [9, 1], 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 [9, 1] 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 [14], 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.

## Citation

NA Technical Report, Mathematical Institute, Oxford University, 2015.