Group-sparsity is a common low-complexity signal model with widespread application across various domains of science and engineering. The recovery of such signal ensembles from compressive measurements has been extensively studied in the literature under the assumption that measurement operators are modeled as densely populated random matrices. In this paper, we turn our attention to an acquisition model intended to ease the energy consumption of sensing devices by splitting the measurements up into distinct signal blocks. More precisely, we present uniform guarantees for group-sparse signal recovery in the scenario where a number of sensors obtain independent partial signal observations modeled by block diagonal measurement matrices. We establish a group-sparse variant of the classical restricted isometry property (RIP) for block diagonal sensing matrices acting on group-sparse vectors, and provide conditions under which subgaussian block diagonal random matrices satisfy this group-RIP with high probability. Two different scenarios are considered in particular. In the first scenario, we assume that each sensor is equipped with an independently drawn measurement matrix. We later lift this requirement by considering measurement matrices with constant block diagonal entries. In other words, every sensor is equipped with a copy of the same prototype matrix. The problem of establishing the group-RIP is cast into a form in which one needs to establish the concentration behavior of the suprema of chaos processes which involves estimating Talagrand's ${\gamma }_2$ functional. As a side effect of the proof, we present an extension to Maurey's empirical method to provide new bounds on the covering number of sets consisting of finite convex combinations of possibly infinite sets.

群稀疏性是一种常见的低复杂度信号模型，在科学和工程的各个领域都有广泛的应用。在假设测量算子被建模为密集分布的随机矩阵的假设下，从压缩测量中恢复此类信号集合已在文献中进行了广泛的研究。在本文中，我们将注意力转向了一种采集模型，该模型旨在通过将测量结果分成不同的信号块来减轻传感设备的能耗。更准确地说，在多个传感器获得由块对角测量矩阵建模的独立部分信号观测值的情况下，我们为组稀疏信号恢复提供了统一保证。我们建立了经典受限等距属性的组稀疏变体（RIP）用于块对角感知矩阵作用于组稀疏向量，并提供亚高斯块对角随机矩阵以高概率满足该组RIP的条件。特别考虑了两种不同的情况。在第一个场景中，我们假设每个传感器都配备了一个独立绘制的测量矩阵。我们稍后通过考虑具有恒定块对角条目的测量矩阵来提高这一要求。换句话说，每个传感器都配备了相同原型矩阵的副本。建立群 RIP 的问题被转化为一种形式，其中需要建立混沌过程至上的集中行为，其中涉及估计 Talagrand 的${\gamma }_2$功能性的。作为证明的一个副作用，我们提出了 Maurey 经验方法的扩展，以提供由可能无限集的有限凸组合组成的集合的覆盖数的新界限。