Iterative Soft/Hard Thresholding with Homotopy Continuation for Sparse Recovery

In this note, we analyze an iterative soft / hard thresholding algorithm with homotopy continuation for recovering a sparse signal $x^\dag$ from noisy data of a noise level $\epsilon$. Under suitable regularity and sparsity conditions, we design a path along which the algorithm can find a solution $x^*$ which admits a sharp reconstruction error $\|x^* - x^\dag\|_{\ell^\infty} = O(\epsilon)$ with an iteration complexity $O(\frac{\ln \epsilon}{\ln \gamma} np)$, where $n$ and $p$ are problem dimensionality and $\gamma\in (0,1)$ controls the length of the path. Numerical examples are given to illustrate its performance.