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This talk discusses the unstructured sparse recovery problems of a general form. The task is to recover the spike locations and weights of an unknown sparse signal from a collection of its unstructured observations. 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. We propose the eigenmatrix construction, a data-driven approach to this problem in one and multidimensional cases. The eigenmatrix turns this non-linear inverse problem into an eigen-decomposition problem with desired eigenvalue and eigenvector pairs.