One-bit compressive sensing with norm estimation

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 $\operatorname{sign}(\langle {a}i, {x} \rangle) \in {\pm1}$. Since such measurements give no information on the norm of ${x}$, recovery methods from such measurements typically assume that $| {x} |2=1$. We show that if one allows more generally for quantized affine measurements of the form $\operatorname{sign}(\langle {a}i, {x} \rangle + bi)$, and if the vectors ${a}i$ are random, an appropriate choice of the affine shifts $bi$ 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 $r \leq | {x} |2 \leq R$, one may estimate the norm $| {x} |2$ up to additive error $\delta$ from $m \gtrsim R⁴ r^{-2} \delta^{-2}$ 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 ${x}$ within a Euclidean ball of known radius.