Estimation of block sparsity in compressive sensing

Abstract

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 $k_{\alpha }(\mathbf{x})={\left(\Vert \mathbf{x}{\Vert }_{2,\alpha }/\Vert \mathbf{x}{\Vert }_{2,1}\right)}^{\frac{\alpha }{1-\alpha }}$ 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.

References in corpus (2)

• Structured Compressed Sensing: From Theory to Applications
• Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors

Cited by in corpus (3)

• Enhanced block sparse signal recovery based on $q$-ratio block constrained minimal singular values
• Sparse recovery based on q-ratio constrained minimal singular values
• Statistical inference for block sparsity of complex signals