Estimation of block sparsity in compressive sensing

Explicitly using the block structure of the unknown signal can achieve better reconstruction performance in compressive sensing. Theoretically, an unknown signal with block structure can be accurately recovered from a few number of under-determined linear measurements provided that it is sufficiently block sparse. From the practical point of view, a severe concern is that the block sparse level appears often unknown. In this paper, we introduce a soft measure of block sparsity $k_\alpha(\mathbf{x})=\left(\lVert\mathbf{x}\rVert_{2,\alpha}/\lVert\mathbf{x}\rVert_{2,1}\right)^{\frac{\alpha}{1-\alpha}}$ with $\alpha\in[0,\infty]$, and propose an estimation procedure by using multivariate centered isotropic symmetric $\alpha$-stable random projections. The limiting distribution of the estimator is established. Simulations are conducted to illustrate our theoretical results.