The problem of finding the sparsest solution to a linear underdetermined system of equations, often appearing, e.g., in data analysis, optimal control, system identification, or sensor selection problems, is considered. This non-convex problem is commonly solved by convexification via $\ell_1$-norm minimization, known as basis pursuit (BP). In this work, a class of structured matrices, representing the system of equations, is introduced for which (BP) tractably fails to recover the sparsest solution. In particular, this enables efficient identification of matrix columns corresponding to unrecoverable non-zero entries of the sparsest solution and determination of the uniqueness of such a solution. These deterministic guarantees complement popular probabilistic ones and provide insights into the a priori design of sparse optimization problems. As our matrix structures appear naturally in optimal control problems, we exemplify our findings based on a fuel-optimal control problem for a class of discrete-time linear time-invariant systems. Finally, we draw connections of our results to compressed sensing and common basis functions in geometric modeling.

翻译：本文研究了为线性欠定方程组寻找最稀疏解的问题，该问题常见于数据分析、最优控制、系统辨识或传感器选择等领域。这一非凸问题通常通过$\ell_1$范数最小化的凸化方法求解，即基追踪（BP）。本研究引入了一类表示方程组的结构化矩阵，对于此类矩阵，（BP）可证明地无法恢复最稀疏解。具体而言，该方法能有效识别与最稀疏解中不可恢复非零项对应的矩阵列，并判定此类解的唯一性。这些确定性保证补充了常见的概率性结论，为稀疏优化问题的先验设计提供了理论依据。由于所提出的矩阵结构天然存在于最优控制问题中，我们以一类离散时间线性时不变系统的燃料最优控制问题为例验证了研究结果。最后，我们将所得结论与压缩感知及几何建模中的常用基函数建立了理论关联。