Robust Compressed Sensing Analysis and Solution Based on  \(\ell _1/\ell _2\) Model

This paper proposes an improved robust compressed sensing (RCS) framework based on the non-convex \(\ell _1/\ell _2\) regularization model, addressing key challenges in computational efficiency and noise resilience. Building on the theoretical foundations of \(\ell _1/\ell _2\) minimization [11, 16], we establish the equivalence of critical points between the constrained and unconstrained formulations, ensuring global convergence via the Kurdyka-Łojasiewicz property. Inspired by parametric methods for fractional optimization [17], we develop first-order algorithms, namely, the proximity-gradient-subgradient-subgradient algorithm (PGSSA) and its line search variant (PGSSA_L). Numerical experiments reveal that compared to the traditional MBA method, the new algorithm improves noise robustness by approximately 10% and computational efficiency by 2.5 times, aligning with advancements in accelerated \(\ell _1/\ell _2\) schemes [13]. Our work extends the RCS paradigm [4] by offering a scalable solution for high-dimensional sparse recovery, validated under Gaussian noise and outlier corruption. The results highlight the potential of non-convex regularization in applications such as medical imaging and wireless communications, where efficiency and robustness are critical.