论文标题
压缩感测数据的反问题的联合解决方案的收敛速率
Convergence rates for the joint solution of inverse problems with compressed sensing data
论文作者
论文摘要
压缩传感（CS）是减少在保持高空间分辨率时收集的数据量的强大工具。这样的技术在实践中效果很好，同时也得到了坚实的理论的支持。标准CS结果假设要直接在目标信号上进行测量。但是，在许多实际应用中，CS信息只能从间接数据中获取$ H_ \ star = W x_ \ star $与原始信号相关的额外远期操作员。如果颠倒向前操作员是不适合的，则现有的CS理论不适用。在本文中，我们解决了这一问题，并提出了两种联合重建方法，即放松$ \ ell^1 $共同注册和严格的$ \ ell^1 $共同指导，用于从间接数据中进行CS。作为主要结果，我们得出了恢复$ x_ \ star $和$ h_ \ star $的错误估计。特别是，我们得出了后者规范的线性收敛速率。为了获得这些结果，需要解决方案来满足源条件，并且需要CS测量操作员以满足受限制的注射率条件。我们进一步表明，这些条件不仅足够，而且甚至是获得线性收敛所必需的。
Compressed sensing (CS) is a powerful tool for reducing the amount of data to be collected while maintaining high spatial resolution. Such techniques work well in practice and at the same time are supported by solid theory. Standard CS results assume measurements to be made directly on the targeted signal. In many practical applications, however, CS information can only be taken from indirect data $h_\star = W x_\star$ related to the original signal by an additional forward operator. If inverting the forward operator is ill-posed, then existing CS theory is not applicable. In this paper, we address this issue and present two joint reconstruction approaches, namely relaxed $\ell^1$ co-regularization and strict $\ell^1$ co-regularization, for CS from indirect data. As main results, we derive error estimates for recovering $x_\star$ and $h_\star$. In particular, we derive a linear convergence rate in the norm for the latter. To obtain these results, solutions are required to satisfy a source condition and the CS measurement operator is required to satisfy a restricted injectivity condition. We further show that these conditions are not only sufficient but even necessary to obtain linear convergence.
