A General Compressive Sensing Construct Using Density Evolution

Hang Zhang, Afshin Abdi, Faramarz Fekri

Abstract: This paper proposes a general framework to design a sparse sensing matrix $\mathbf {A}\in \mathbb{R}^{m\times n}$, for a linear measurement system $\mathbf {y} = \mathbf {Ax}^{\natural } + \mathbf {w}$, where $\mathbf {y} \in \mathbb{R}^{m}$, $\mathbf {x}^{\natural }\in \mathbb{R}^{n}$, and $\mathbf {w}\in \mathbb{R}^{m}$ 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, namely the density evolution technique, and provide a framework for the design of matrix $\mathbf {A}$. Two design schemes for the sensing matrix, namely, $(i)$ a regular sensing and $(ii)$ a preferential sensing, are proposed and are incorporated into a single framework. As illustration, we consider the Lasso regression, ridge regression, and elastic net regression; and show that our framework can reproduce the classical results on the minimum sensor number, i.e., $m$. In the preferential sensing scenario, we consider the case in which the whole signal is divided into two disjoint parts, namely, high-priority part $\boldsymbol{x}^{\natural }_{\mathsf {H}}$ and low-priority part $\boldsymbol{x}^{\natural }_{\mathsf {L}}$. Then, by formulating the sensing system design as a bi-convex optimization problem, we obtain sensing matrices which can provide a preferential treatment for $\boldsymbol{x}^{\natural }_{\mathsf {H}}$. Numerical experiments with both synthetic data and real-world data are also provided to verify the effectiveness of our design scheme.

External IDs:dblp:journals/tsp/ZhangAF24