In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. [26] (Group Projected subspace pursuit for IDENTification of variable coefficient differential equations (GP-IDENT), Journal of Computational Physics, 494, 112526) and extend its application to general tasks of block sparse signal recovery. Given an observation y and sampling matrix A, we focus on minimizing the approximation error ‖Ac−y‖22 with respect to the signal c with block sparsity constraints. We prove that when the sampling matrix A satisfies the Block Restricted Isometry Property (BRIP) with a sufficiently small Block Restricted Isometry Constant (BRIC), GPSP exactly recovers the true block sparse signals. When the observations are noisy, this convergence property of GPSP remains valid if the magnitude of the true signal is sufficiently large. GPSP selects the features by subspace projection criterion (SPC) for candidate inclusion and response magnitude criterion (RMC) for candidate exclusion. We compare these criteria with counterparts of other state-of-the-art greedy algorithms. Our theoretical analysis and numerical ablation studies reveal that SPC is critical to the superior performances of GPSP, and that RMC can enhance the robustness of feature identification when observations contain noises. We test and compare GPSP with other methods in diverse settings, including heterogeneous random block matrices, inexact observations, face recognition, and PDE identification. We find that GPSP outperforms the other algorithms in most cases for various levels of block sparsity and block sizes, justifying its effectiveness for general applications.

中文翻译：

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面向块稀疏信号重构的群投影子空间追求：收敛分析与应用


在本文中，我们提出了 He 等人 [26] 提出的群投影子空间追踪 （GPSP） 算法的收敛分析（用于可变系数微分方程 IDENTification 的群投影子空间追踪 （GP-IDENT），计算物理学杂志，494,112526），并将其应用扩展到块稀疏信号恢复的一般任务。给定一个观测值 y 和采样矩阵 A，我们专注于最小化关于具有块稀疏性约束的信号 c 的近似误差 ‖Ac−y‖22。我们证明，当采样矩阵 A 满足块限制等距属性 （BRIP） 和足够小的块限制等距常数 （BRIC） 时，GPSP 可以准确地恢复真正的块稀疏信号。当观测值有噪声时，如果真实信号的幅度足够大，则 GPSP 的这种收敛特性仍然有效。GPSP 通过子空间投影标准 （SPC） 选择特征进行候选纳入，通过响应幅度标准 （RMC） 选择特征进行候选排除。我们将这些标准与其他最先进的贪婪算法进行比较。我们的理论分析和数值消融研究表明，SPC 对于 GPSP 的卓越性能至关重要，当观测包含噪声时，RMC 可以提高特征识别的鲁棒性。我们在不同环境中测试 GPSP 并与其他方法进行比较，包括异构随机块矩阵、不精确观察、人脸识别和 PDE 识别。我们发现，在大多数情况下，GPSP 在各种级别的块稀疏性和块大小方面优于其他算法，证明了它对一般应用程序的有效性。