从间接观测中恢复稀疏信号的线性像

中文摘要 AI 辅助

在本文中，我们开发并分析了从间接含噪观测 $\omega=Ax+\xi$ 中恢复未知信号 $x$ 的线性像 $Bx$ 的技术。先验已知 $x\in\cX$，其中 $\cX$ 是给定的凸紧集，且 $x$ 是 $s$-稀疏的——即至多有 $s$ 个非零元素。所提出的估计属于一大类基于 $\ell_1$-最小化的恢复方法。然而，与描述此类估计性能的经典结果不同，我们不对感知矩阵 $A$ 做出任何特殊（且难以验证）的假设，如零空间条件或受限等距性质等。因此，估计的参数及其风险上界没有闭合解析形式，而是通过高效计算作为显式凸优化问题的解来获得。

英文摘要

In this paper, we develop and analyze techniques for recovering a linear image $Bx$ of an unknown signal $x$ from indirect noisy observation $ω=Ax+ξ$. It is {\em a priori} known that $x\in \cX$, a given convex compact set, and that $x$ is $s$-sparse---has at most $s$ nonvanishing entries. The proposed estimates belong to a large family of recovery routines by $\ell_1$-minimization. However, unlike the classical result describing performance of such estimates, we do not make any special (and hard to check) assumptions about the sensing matrix $A$ such as nullspace or Restricted Isometry condition and the like. As a consequence, parameters of the estimates and the upper bounds on their risks are not available in a closed analytic form, but are delivered instead by efficient computation as solutions to explicit convex optimization problems.

发表机构

• Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）
• Georgia Institute of Technology（佐治亚理工学院）

机构由 AI 辅助整理，请以论文原文为准。