One-Bit Compressive Sensing With Norm Estimation

Abstract

Consider the recovery of an unknown signal x from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that x is sparse, and that the measurements are of the form sign( a i , x ) ∈ {±1}. Since such measurements give no information on the norm of x, recovery methods typically assume that x 2 = 1. We show that if one allows more generally for quantized affine measurements of the form sign( a i , x + b i ), and if the vectors a i are random, an appropriate choice of the affine shifts b i allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. Additionally, we show that for arbitrary fixed x in the annulus r ≤ x 2 ≤ R, one may estimate the norm x 2 up to additive error δ from m R 4 r −2 δ −2 such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors, in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors x in a Euclidean ball of known radius.

Quantized Massive MU-MIMO-OFDM Uplink

“…There, it was shown that amplitude information about the transmitted signal can be recovered even in the presence of a single-bit quantizer, provided that the number of receive antennas is sufficiently large and the signalto-noise ratio (SNR) is not too high. The observation that noise can help recovering magnitude information under 1-bit quantization has also been made in the compressive-sensing literature [9], [10].…”

Dense Quantum Measurement Theory

“…A special technique within compressed sensing is the so-called "1-bit" compressed sensing [68][69][70], where 1-bit measurements are applied that preserve only the sign information of the measurements.…”

Stronger L 2 /L 2 compressed sensing; without iterating

“…The thresholding step, namely the part where we take use V to filter out non-heavy intervals cannot be implemented here, but perhaps there is still a way to make a similar argument. One first approach should be to show that sublinear decoding with optimal measurements is achieved using non-adaptive threshold measurements, such as in [KSW16] and [BFN + 17] (note that the latter one uses adaptive measurements though).…”