论文标题
使用密度演化的一般压缩传感构建体
A General Compressive Sensing Construct using Density Evolution
论文作者
论文摘要
本文提出了一个通用框架，以设计稀疏的传感矩阵$ \ ensureMath {\ Mathbf {a}} \ in \ Mathbb {r}^{r}^{m \ times n} $，在线性测量系统$ \ ensureMath $ \ ensureMath {\ mathbf {y MathBf {y Mathbf {y Mathbf {y Mathbf {Y} \ ensureMath {\ MathBf {ax}}}^{\ natural} + \ ensureMath {\ mathbf {w}} $，其中$ \ ensureMath {\ senuremath {\ mathbf {y}} \ $ \ ensureMath {\ Mathbf {x}}^{\ natural} \ in \ rr^n $，以及$ \ ensureMath {\ MathBf {w}} $分别表示测量，信号，某些结构和测量噪声。通过将测量值的信号重构视为通过图形模型传递算法传递算法的消息，我们利用编码理论从编码理论的设计中，在低密度平价检查代码的设计中，即密度演化，并为矩阵$ \ ensuremath {\ semuremath {\ nathbf {a}} $提供设计框架。特别是，与以前的方法相比，我们提出的框架享有以下理想的属性：（$ i $）通用性：该设计支持常规感测和优先感应，并将它们整合到一个框架中； （$ ii $）灵活性：该框架可以轻松地将$ \ ba $的设计适应信号$ \ ensureMath {\ Mathbf {x}}}}^{\ natural} $具有不同的基础结构。作为例证，我们考虑与常规感测和优先传感方案相对应的$ \ ell_1 $正规器。值得注意的是，我们的框架可以重现拉索的经典结果，即$ m \ geq c_0 k \ log（n/k）$（常规感应），具有正常设计后的定期设计，其中$ c_0> 0 $是固定的常数。我们还提供数值实验来确认分析结果，并在需要对矢量$ \ bx^{\ natural} $的优先处理时证明我们的框架的优越性。
This paper proposes a general framework to design a sparse sensing matrix $\ensuremath{\mathbf{A}}\in \mathbb{R}^{m\times n}$, in a linear measurement system $\ensuremath{\mathbf{y}} = \ensuremath{\mathbf{Ax}}^{\natural} + \ensuremath{\mathbf{w}}$, where $\ensuremath{\mathbf{y}} \in \mathbb{R}^m$, $\ensuremath{\mathbf{x}}^{\natural}\in \RR^n$, and $\ensuremath{\mathbf{w}}$ denote the measurements, the signal with certain structures, and the measurement noise, respectively. By viewing the signal reconstruction from the measurements as a message passing algorithm over a graphical model, we leverage tools from coding theory in the design of low density parity check codes, namely the density evolution, and provide a framework for the design of matrix $\ensuremath{\mathbf{A}}$. Particularly, compared to the previous methods, our proposed framework enjoys the following desirable properties: ($i$) Universality: the design supports both regular sensing and preferential sensing, and incorporates them in a single framework; ($ii$) Flexibility: the framework can easily adapt the design of $\bA$ to a signal $\ensuremath{\mathbf{x}}^{\natural}$ with different underlying structures. As an illustration, we consider the $\ell_1$ regularizer, which correspond to Lasso, for both the regular sensing and preferential sensing scheme. Noteworthy, our framework can reproduce the classical result of Lasso, i.e., $m\geq c_0 k\log(n/k)$ (the regular sensing) with regular design after proper distribution approximation, where $c_0 > 0$ is some fixed constant. We also provide numerical experiments to confirm the analytical results and demonstrate the superiority of our framework whenever a preferential treatment of a sub-block of vector $\bx^{\natural}$ is required.
