Sparse recovery aims to reconstruct sparse signals from underdetermined and possibly noisy linear measurements. Existing $\ell_{1-2}$ iterative thresholding schemes are first-order methods. We propose an iterative thresholding pursuit method with continuation (ITP-C) for $\ell_{1-2}$-regularized sparse recovery. The method goes beyond first-order thresholding by combining the active-set identification capability of the $\ell_{1-2}$ proximal step with a restricted least-squares pursuit step that provides a second-order update on the identified support. The support is generated adaptively by the thresholding update, and no prior knowledge of the true sparsity level is required. To control the possible instability of the pursuit step while preserving the descent structure of the continuation scheme, we impose a strict descent check with respect to the dynamic objective. We establish convergence of the generated sequence under the Kurdyka-Lojasiewicz framework and prove a local oracle-type property after correct support identification. Numerical experiments on synthetic sparse recovery and image reconstruction illustrate the descent preservation of the proposed safeguard and demonstrate the improved recovery performance of ITP-C over the state-of-the-art baselines.

翻译：稀疏恢复旨在从欠定且可能含噪声的线性测量中重构稀疏信号。现有的$ \ell_{1-2} $迭代阈值方案属于一阶方法。我们提出了一种带连续策略的迭代阈值追踪方法（ITP-C）用于$ \ell_{1-2} $正则化稀疏恢复。该方法结合了$ \ell_{1-2} $近端步的主动集识别能力与限制性最小二乘追踪步（提供已识别支撑上的二阶更新），从而超越了一阶阈值方法的范畴。支撑集通过阈值更新自适应生成，无需预先了解真实稀疏度水平。为控制追踪步可能引起的不稳定性同时保留连续策略的下降结构，我们针对动态目标函数施加了严格的下降检查。在Kurdyka-Lojasiewicz框架下建立了生成序列的收敛性，并证明了正确支撑识别后的局部预言机型性质。针对合成稀疏恢复与图像重建的数值实验表明，所提保护机制保持了下降特性，并展示了ITP-C方法相较于最先进基准在恢复性能上的显著提升。