In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: y = η ⊙ sign(Ψx * + ǫ), where x * ∈ R n , y ∈ R m , Ψ ∈ R m×n , and ǫ is the random error before quantization and η ∈ R n is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined settings. For m > n, we show that, up to a constant c, with high probability, the least squares solution x ls approximates x * with precision δ as long as m ≥ O( n δ 2 ). For m < n, we prove that, up to a constant c, with high probability, the ℓ 1 -regularized least-squares solution x ℓ 1 lies in the ball with center x * and radius δ provided that m ≥ O( s log n δ 2 ) and x * 0 := s < m. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm.Keywords: 1-bit compressive sensing, ℓ 1 -regularized least squares, primal dual active set algorithm, one step convergence, continuation 1. Introduction. Compressive sensing (CS) is an important approach to acquiring low dimension signals from noisy under-determined measurements [8,16,19,20]. For storage and transmission, the infinite-precision measurements are often quantized, [6] considered recovering the signals from the 1-bit compressive sensing (1-bit CS) where measurements are coded into a single bit, i.e., their signs. The 1-bit CS is superior to the CS in terms of inexpensive hardware implementation and storage. However, it is much more challenging to decode from nonlinear, noisy and sign-flipped 1-bit measurements.1.1. Previous work. Since the seminal work of [6], much effort has been devoted to studying the theoretical and computational challenges of the 1-bit CS. Sample complexity was analyzed for support and vector recovery with and without noise [21,28,40,23,29,22,23,41,50]. Existing works indicate that, m > O(s log n) is adequate for both support and vector recovery. The sample size required here has the same order as that required in the standard CS setting. These results have also been refined by adaptive sampling [22,14,4]. Extensions include recovering the norm of the target [32, 3] and non-Gaussian measurement settings [1]. Many first order methods [6,34,49,14] and greedy methods [35,5,29] are developed to minimize the sparsity promoti...