论文标题
通用的1位压缩传感，用于有限的动态范围信号
Universal 1-Bit Compressive Sensing for Bounded Dynamic Range Signals
论文作者
论文摘要
一个{\ em通用1位压缩传感（CS）}方案由一个测量矩阵$ a $组成，使得所有信号$ x $都可以从$ \ textrm {sign}（ax）$中大约恢复。 1位CS模型的极端量化效果，其中仅显示一个测量的信息。在{\ em稀疏}信号的情况下，我们专注于1位CS的通用支持恢复，并具有有界{\ em dynamic范围}的信号。具体而言，如果最多有$ k $ nonzero条目，则据说vector $ x \ in \ mathbb {r}^n $具有稀疏$ k $，而动态范围$ r $如果其最大和最小的非零条目之间的比率最多是$ r $ $ $。我们的主要结果表明，如果测量矩阵$ a $的条目是i.i.d.〜gaussians，则在对$ k $和$ r $的缩放范围内的温和假设下，测量的数量需要为$ \tildeΩ（rk^{3/2}）$，以恢复$ k $ -sparse $ $ $ $ $ r的支持，以恢复$ k $ -s的支持。此外，我们表明近匹配的$ O（r k^{3/2} \ log n）$上限是已知结果的简单推论。 $ k^{3/2} $缩放与$ \tildeΩ（k^2 \ log n）$的已知下限进行对比，用于恢复任意$ k $ -sparse信号的支持的测量数。
A {\em universal 1-bit compressive sensing (CS)} scheme consists of a measurement matrix $A$ such that all signals $x$ belonging to a particular class 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 $\tildeΩ(Rk^{3/2})$ 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 $\tildeΩ(k^2 \log n)$ for the number of measurements to recover the support of arbitrary $k$-sparse signals.
