  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.
