A {\em universal 1-bit compressive sensing (CS)} scheme consists of a measurement matrix $A$ such that for all signals $x$ belonging to a particular class, $x$ can be approximately recovered from $\textrm{sign}(Ax)$. 1-bit CS models extreme quantization effects where only one bit of information is revealed per measurement. We focus on universal support recovery for 1-bit CS in the case of {\em sparse} signals with bounded {\em dynamic range}. Specifically, a vector $x \in \mathbb{R}^n$ is said to have sparsity $k$ if it has at most $k$ nonzero entries, and dynamic range $R$ if the ratio between its largest and smallest nonzero entries is at most $R$ in magnitude. Our main result shows that if the entries of the measurement matrix $A$ are i.i.d.~Gaussians, then under mild assumptions on the scaling of $k$ and $R$, the number of measurements needs to be $\tilde{\Omega}(Rk^{3/2})$ to recover the support of $k$-sparse signals with dynamic range $R$ using $1$-bit CS. In addition, we show that a near-matching $O(R k^{3/2} \log n)$ upper bound follows as a simple corollary of known results. The $k^{3/2}$ scaling contrasts with the known lower bound of $\tilde{\Omega}(k^2 \log n)$ for the number of measurements to recover the support of arbitrary $k$-sparse signals.
翻译:$1-bit CS 模型的极端量化效应,每次测量只显示一小部分信息。我们的重点是在有限制的动态范围的信号中为1位CS提供普遍支持。具体地说,对于属于某一类的所有信号来说,对于属于某一类的所有信号来说,美元xx $x=mathbb{R ⁇ n$ 据说,美元可以大约从$\textrm{sign}(xxx$xxx$x$x$xxx$xxx$xmathb{R ⁇ n$有螺旋度的对比度。如果最大和非最小的非零条目之间的比例在数量中最多为$R$。我们的主要结果显示,如果测量矩阵的条目为 $A. d.~ gussians,然后根据简单的假设 美元=k_xxxxxxxxxxxxmaxxxxxmathr_R_r_r_xxxxxxxxxxxxxxxxxxxal_r_r_r_r_rmaxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx