One-bit compressed sensing via ℓp(0 < p < 1)-minimization method

Abstract

One-bit compressed sensing aims to recover unknown sparse signals from extremely quantized linear measurements which just capture their signs. In this paper, we propose a nonconvex ℓp(0 < p < 1) minimization model for one-bit compressed sensing problem and define the set of ℓp effectively s-sparse signals which contains genuinely s-sparse signals. Utilizing properties of covering number, we show that our method can recover the direction of ℓp effectively s-sparse signals with error order . We also employ thresholded one-bit measurements to estimate the magnitude of signals and prove that any ℓp effectively s-sparse bounded signal x can be estimated using augmented ℓp minimization model and empirical distribution function method respectively. Especially, to recover ℓp effectively s-sparse signals in practice, we introduce an adaptive binary iterative thresholding algorithm which can be utilized without knowing the sparsity of underlying signals. Numerical experiments on both synthetic and real-world data sets are conducted to demonstrate the superiority of our algorithm.