In this paper, we introduce the q-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed ℓ₂/ℓ_q and the mixed ℓ₂/ℓ₁ norms of the reconstruction errors for stable and robust recovery using block basis pursuit (BBP), the block Dantzig selector (BDS), and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise-free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV-based error bounds are tighter than the block-restricted isotropic constant-based bounds.

在本文中，我们介绍了q比率块约束最小奇异值（BCMSV）作为块稀疏/压缩信号压缩感知中测量矩阵的一种新度量，并提出了一种计算该新度量的算法。二者混合ℓ ₂ / ℓ _q和混合ℓ ₂ / ℓ _1个规范进行稳定和稳健恢复使用块为基础追求重建误差（BBP），块丹选择（BDS）和条款套索的基团的研究q比BCMSV。我们根据q建立充分条件-比块稀疏性，以从无噪声BBP中精确恢复，并开发了一个凸凹过程来解决该条件下的相应非凸问题。此外，我们证明了对于次高斯随机矩阵，当测量数量相当大时，q比率BCMSV很有可能远离零。进行了数值实验以说明理论结果。此外，我们证明了基于q比率BCMSV的错误界限比基于块限制的各向同性常数的界限更严格。