This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. This note proposes the eigenmatrix, a data-driven construction with desired approximate eigenvalues and eigenvectors. The eigenmatrix offers a new way for these sparse recovery problems. Numerical results are provided to demonstrate the efficiency of the proposed method.

翻译：本文考虑一般形式的非结构化稀疏恢复问题，其示例包括有理逼近、谱函数估计、傅里叶反演、拉普拉斯反演以及稀疏反卷积。主要挑战在于样本值中的噪声以及样本位置的非结构化特性。本文提出特征矩阵，这是一种具有所需近似特征值和特征向量的数据驱动型构造。特征矩阵为这些稀疏恢复问题提供了一种新途径。数值结果证明了所提方法的高效性。