Quantitative Recovery Conditions for Tree-Based Compressed Sensing
IEEE Transactions on Information Theory
As shown in ,  , signals whose wavelet coefficients exhibit a rooted tree structure can be recovered using speciallyadapted compressed sensing algorithms from just n = 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
... is 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, recently introduced in , 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 to 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 ≥ 50k. All our results extend to the more realistic case in which measurements are corrupted by noise.