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^† from noisy data of a noise level ε. 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^†|| ℓ∞ = O(ε) with an iteration complexity O((ln ε)/(ln γ)np), where n and p are problem dimensionality and γ ε (0,1) controls the length of the path. Numerical examples are given to illustrate its performance.