Nearly Linear-Time Model-Based Compressive Sensing

Chinmay Hegde, Piotr Indyk, Ludwig Schmidt

Abstract: Compressive sensing is a method for recording a k-sparse signal x ∈ ℝn with (possibly noisy) linear measurements of the form y = Ax, where A ∈ ℝm ×n describes the measurement process. Seminal results in compressive sensing show that it is possible to recover the signal x from \(m = O(k \log \frac{n}{k})\) measurements and that this is tight. The model-based compressive sensing framework overcomes this lower bound and reduces the number of measurements further to m = O(k). This improvement is achieved by limiting the supports of x to a structured sparsity model, which is a subset of all \(\binom{n}{k}\) possible k-sparse supports. This approach has led to measurement-efficient recovery schemes for a variety of signal models, including tree-sparsity and block-sparsity.