Robust Sparse Signal Recovery with Outliers: A Hard Thresholding Pursuit Approach Based on LAD

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.