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:在壓縮測量中恢復稀疏信號的理論條件已經得到了廣泛的研究。與此同時,還有大量的工作研究了線性動態系統 (LDS) 的狀態和輸入恢復,包括一些關於恢復稀疏輸入的結果。然而,現有的恢復LDS稀疏輸入的充分條件是保守且難以解釋的,而且尚未在文獻中提出必要且充分的條件。在本文中,我們提供了如下貢獻:(1) 闡明了LDS稀疏輸入的存在性和唯一性的必要和充分條件,(2) 提供了線性規劃可恢復未知初始狀態和稀疏輸入的必要和充分條件,和(3) 以LDS參數簡單且易於解釋的方式進行恢復。最後,我們通過數值實驗驗證了這些結論,並討論了相關影響和未來的研究方向。