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，包括有理逼近、谱函数估计、傅里叶逆变换、拉普拉斯逆变换以及稀疏反卷积等典型实例。主要挑战在于样本值的噪声特性以及采样位置的非结构化特征。本文提出一种数据驱动的特征矩阵构造方法，该矩阵具有期望的近似特征值和特征向量。特征矩阵为这些稀疏恢复问题提供了全新解决途径。数值实验结果验证了所提方法的有效性。