In this paper, we present a convergence analysis of the Group Projected Subspace Pursuit (GPSP) algorithm proposed by He et al. [HKL+23] (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. We prove that when the sampling matrix 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 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.

翻译：本文对He等人提出的群组投影子空间追踪（GPSP）算法（HKL+23，《变系数微分方程识别的群组投影子空间追踪（GP-IDENT）》，《计算物理学杂志》，第494卷，第112526页）进行了收敛性分析，并将其应用扩展至块稀疏信号恢复的一般任务。我们证明，当采样矩阵满足具有足够小块限制等距常数（BRIC）的块限制等距性质（BRIP）时，GPSP能够精确恢复真实的块稀疏信号。在观测数据存在噪声的情况下，若真实信号的幅值足够大，GPSP的收敛性质依然成立。GPSP通过子空间投影准则（SPC）进行候选特征纳入，并通过响应幅值准则（RMC）进行候选特征排除。我们将这些准则与其他先进贪婪算法的对应机制进行了比较。理论分析与数值消融研究表明，SPC对GPSP的优越性能至关重要，而RMC能在观测含噪声时增强特征识别的鲁棒性。我们在多种场景下测试并比较了GPSP与其他方法，包括异构随机块矩阵、非精确观测、人脸识别及偏微分方程识别。研究发现，在不同块稀疏度与块尺寸条件下，GPSP在多数情况下优于其他算法，这证实了其在通用应用场景中的有效性。