The study of theoretical conditions for recovering sparse signals from compressive measurements has received a lot of attention in the research community. In parallel, there has been a great amount of work characterizing conditions for the recovery both the state and the input to a linear dynamical system (LDS), including a handful of results on recovering sparse inputs. However, existing sufficient conditions for recovering sparse inputs to an LDS are conservative and hard to interpret, while necessary and sufficient conditions have not yet appeared in the literature. In this work, we provide (1) the first characterization of necessary and sufficient conditions for the existence and uniqueness of sparse inputs to an LDS, (2) the first necessary and sufficient conditions for a linear program to recover both an unknown initial state and a sparse input, and (3) simple, interpretable recovery conditions in terms of the LDS parameters. We conclude with a numerical validation of these claims and discuss implications and future directions.

翻译：利用压缩测量恢复稀疏信号的理论条件研究已受到学术界的广泛关注。与此同时，大量研究致力于刻画线性动力系统(LDS)状态与输入同时恢复的条件，其中包括若干关于稀疏输入恢复的成果。然而，现有关于LDS稀疏输入恢复的充分条件过于保守且难以解释，而充要条件尚未在文献中出现。本文提出了：(1) LDS稀疏输入存在性与唯一性的首个充要条件刻画；(2) 线性规划同时恢复未知初始状态与稀疏输入的首个充要条件；(3) 基于LDS参数的简洁可解释恢复条件。最后通过数值验证这些结论，并讨论其影响与未来方向。