A simple homotopy proximal mapping algorithm for compressive sensing

Abstract

In this paper, we present novel yet simple homotopy proximal mapping algorithms for reconstructing a sparse signal from (noisy) linear measurements of the signal or for learning a sparse linear model from observed data, where the former task is well-known in the field of compressive sensing and the latter task is known as model selection in statistics and machine learning. The algorithms adopt a simple proximal mapping of the $$\ell _1$$ norm at each iteration and gradually reduces the regularization parameter for the $$\ell _1$$ norm. We prove a global linear convergence of the proposed homotopy proximal mapping (HPM) algorithms for recovering the sparse signal under three different settings (i) sparse signal recovery under noiseless measurements, (ii) sparse signal recovery under noisy measurements, and (iii) nearly-sparse signal recovery under sub-Gaussian noisy measurements. In particular, we show that when the measurement matrix satisfies restricted isometric properties (RIP), one of the proposed algorithms with an appropriate setting of a parameter based on the RIP constants converges linearly to the optimal solution up to the noise level. In addition, in setting (iii), a practical variant of the proposed algorithms does not rely on the RIP constants and our results for sparse signal recovery are better than the previous results in the sense that our recovery error bound is smaller. Furthermore, our analysis explicitly exhibits that more observations lead to not only more accurate recovery but also faster convergence. Finally our empirical studies provide further support for the proposed homotopy proximal mapping algorithm and verify the theoretical results.