Generalized Vandermonde decomposition and its use for multi-dimensional super-resolution

The Vandermonde decomposition of Toeplitz matrices, discovered by Carathéodory and Fejér in the 1910s and rediscovered by Pisarenko in the 1970s, forms the basis of modern subspace methods for 1D frequency estimation. Many related numerical tools have also been developed for multi-dimensional (MD), especially 2D, frequency estimation; however, a fundamental question has remained unresolved as to whether an analog of the Vandermonde decomposition holds in the MD case. In this paper, an affirmative answer to this question and a constructive method for finding the decomposition are provided under appropriate conditions. The new result is also used to study MD frequency estimation from compressive data within the recent super-resolution framework. A systematic approach is proposed and a numerical simulation is provided to demonstrate its effectiveness compared to the existing atomic norm method.