  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 describe measurement mappings achieving, with
overwhelming probability, nearly optimal error decay. Next, we consider
reconstruction robustness to measurement errors and noise and introduce the
Binary $\epsilon$-Stable Embedding (B$\epsilon$SE) property, which
characterizes the robustness measurement process to sign changes. We show 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.
