The Compressive Sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit-depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices provide measurement mappings that, with overwhelming probability, achieve nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the Binary -Stable Embedding (B SE) property, which characterizes the robustness of the measurement process to sign changes. We show that the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the Binary Iterative Hard Thresholding (BIHT) algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.

Abstract-Compressive sensing is a new signal acquisition technology with the potential to reduce the number of measurements required to acquire signals that are sparse or compressible in some basis. Rather than uniformly sampling the signal, compressive sensing computes inner products with a randomized dictionary of test functions. The signal is then recovered by a convex optimization that ensures the recovered signal is both consistent with the measurements and sparse. Compressive sensing reconstruction has been shown to be robust to multi-level quantization of the measurements, in which the reconstruction algorithm is modified to recover a sparse signal consistent to the quantization measurements. In this paper we consider the limiting case of 1-bit measurements, which preserve only the sign information of the random measurements. Although it is possible to reconstruct using the classical compressive sensing approach by treating the 1-bit measurements as ±1 measurement values, in this paper we reformulate the problem by treating the 1-bit measurements as sign constraints and further constraining the optimization to recover a signal on the unit sphere. Thus the sparse signal is recovered within a scaling factor. We demonstrate that this approach performs significantly better compared to the classical compressive sensing reconstruction methods, even as the signal becomes less sparse and as the number of measurements increases.

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Abstract-Finding optimal sparse solutions to estimation problems, particularly in underdetermined regimes has recently gained much attention. Most existing literature study linear models in which the squared error is used as the measure of discrepancy to be minimized. However, in many applications discrepancy is measured in more general forms such as loglikelihood. Regularization by 1-norm has been shown to induce sparse solutions, but their sparsity level can be merely suboptimal. In this paper we present a greedy algorithm, dubbed Gradient Support Pursuit (GraSP), for sparsity-constrained optimization. Quantifiable guarantees are provided for GraSP when cost functions have the "Stable Hessian Property".

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