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
$\operatorname{sign}(\langle {a}_i, {x} \rangle + b_i)$, 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 $r \leq \| {x} \|_2 \leq R$, one may estimate the norm $\| {x}
\|_2$ up to additive error $\delta$ from $m \gtrsim R^4 r^{-2} \delta^{-2}$
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.

在本文中，我们探讨了量化线性测量的恢复问题，提出使用量化仿射测量可更好地保留信号的范数信息，并在一定条件下更容易实现稀疏信号的恢复，并可以在已知半径的欧几里得球内成功地估计所有这些稀疏向量的范数。