Convex Reconstruction of Structured Matrix Signals from Linear Measurements (I): Theoretical Results

Abstract

We investigate the problem of reconstructing n-by-n structured matrix signal X via convex programming, where each column xj is a vector of s-sparsity and all columns have the same l1-norm. The regularizer in use is matrix norm |||X|||1=maxj|xj|1.The contribution in this paper has two parts. The first part is about conditions for stability and robustness in signal reconstruction via solving the convex programming from noise-free or noisy measurements.We establish uniform sufficient conditions which are very close to necessary conditions and non-uniform conditions are also discussed. Similar as the traditional compressive sensing theory for reconstructing vector signals, a related RIP condition is established. In addition, stronger conditions are investigated to guarantee the reconstructed signal's support stability, sign stability and approximation-error robustness. The second part is to establish upper and lower bounds on number of measurements for robust reconstruction in noise. We take the convex geometric approach in random measurement setting and one of the critical ingredients in this approach is to estimate the related widths bounds in case of Gaussian and non-Gaussian distributions. These bounds are explicitly controlled by signal's structural parameters r and s which determine matrix signal's column-wise sparsity and l1-column-flatness respectively.