Recovering a sparse signal from outlier-contaminated measurements is a fundamental challenge in many applications. While existing algorithms predominantly address scenarios with bounded noise or assume known signal sparsity, few methods tackle the more practical problem of sparse recovery from gross outliers without prior knowledge of sparsity. To bridge this gap, we study the sparsity-constrained Least Absolute Deviations (LAD) minimization problem. This paper proposes the Graded Fast Hard Thresholding Pursuit (GFHTP$_1$) algorithm with a quantile-truncated step size for $\ell_1$-loss minimization. In contrast to most state-of-the-art methods, our GFHTP$_1$ requires no prior knowledge of the signal's sparsity level. We establish a theoretical convergence analysis under mild conditions and further prove that an $s$-sparse signal can be recovered exactly within at most $s$ iterations. To our knowledge, these results provide the first efficient recovery guarantees for sparse signal reconstruction from outlier-corrupted measurements without a sparsity prior. Numerical experiments demonstrate that GFHTP$_1$ consistently outperforms competing algorithms in robustness to varying signal sparsity and outlier support size, while also achieving less computational time.

翻译：从受异常值污染的测量中恢复稀疏信号是许多应用中的基本挑战。虽然现有算法主要处理有界噪声场景或假设已知信号稀疏性，但很少有方法能在没有先验稀疏性知识的情况下，从严重异常值中恢复稀疏信号这一更实际的问题。为填补这一空白，我们研究了稀疏约束的最小绝对偏差（LAD）最小化问题。本文提出了用于ℓ₁损失最小化的、具有分位数截断步长的分级快速硬阈值追踪（GFHTP₁）算法。与大多数先进方法不同，我们的GFHTP₁算法无需信号稀疏度水平的先验知识。我们在温和条件下建立了理论收敛性分析，并进一步证明一个s-稀疏信号最多可在s次迭代内被精确恢复。据我们所知，这些结果为在无稀疏性先验条件下从异常值破坏的测量中重建稀疏信号提供了首个高效恢复保证。数值实验表明，GFHTP₁在不同信号稀疏度和异常值支撑集大小的鲁棒性方面始终优于竞争算法，同时计算时间也更少。