Noisy 1-bit compressive sensing: models and algorithms

The compressive sensing (CS) method allows us to recover a sparse signal from a small number of its linear measurements relative to the dimension of the signal space. The classic CS method assumes the measurements to have infinite bit precisions, which cannot be satisfied in practice. Quantization of the measurements is a crucial issue in CS. An extreme quantization, which preserves the signs of the measurements only, attracted much attention recently. The corresponding CS problem is called the 1-bit CS problem since each measurement is quantized to 1 bit. Many models and algorithms were proposed for solving the 1-bit CS problem. Some of them work well when there is no noise (sign flips) in the measurements and a recently proposed algorithm works well when the noise level is known. However, the noise level is not always known in practice. In this paper, we propose a robust one-sided ℓ0 objective model for the 1-bit CS problem by applying the maximum a posteriori (MAP) estimation. In the proposed model, the noise factor is considered without knowing the noise level. We also provide an upper bound of the reconstruction error for the proposed model. To solve the proposed model, we first approximate the objective function of the one-sided ℓ0 model by a continuous function utilizing the Moreau envelope. This leads to an approximate model whose objective function is non-smooth and non-convex. We prove that such an objective function has a finite number of local minimizers. We develop a system of fixed-point equations whose solutions are local minimizers of the approximate model and show that global minimizers of the approximate model are solutions of the fixed-point equations, based on which we propose a fixed-point proximity algorithm for finding local minimizers of the approximate model. We prove that the sequence generated from the algorithm converges to a local minimizer of the approximate model and the sequence of the corresponding objective function values is non-increasing and converges. Furthermore, we prove that the sequence generated from the algorithm converges to a global minimizer of the approximate model as long as the initial guess is sufficiently close to a global minimizer of the approximate model. Numerical results are presented to demonstrate the accuracy of the proposed method.