Quantitative recovery conditions for tree-based compressed sensing

TL;DR

This paper introduces a new framework for quantitatively analyzing the recovery guarantees of tree-based compressed sensing algorithms, providing explicit measurement bounds and improvements over previous worst-case analyses.

Contribution

It develops a proportional-dimensional asymptotic framework and applies it to existing algorithms, yielding explicit measurement bounds and improved average-case guarantees.

Findings

Exact recovery of binary tree-based signals is guaranteed for n>50k in noiseless Gaussian measurements.

The new analysis provides explicit bounds on the number of measurements needed for recovery.

Results extend to noisy measurement scenarios, enhancing practical applicability.

Abstract

As shown in [Blumensath and 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=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, recently introduced in [Cartis and Thompson, 2015], 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.