This paper 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 paper 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，实例包括有理逼近、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。本文提出了一种数据驱动的特征矩阵构造方法，该方法具备所需的近似特征值和特征向量。该特征矩阵为这些稀疏恢复问题提供了一种新途径。文中给出了数值结果，以证明所提方法的有效性。