Fast Kernel-based Signal Subspace Estimates for Line Spectral Estimation

This paper introduces the problem formulation of kernel-based subspace estimates for line spectral estimation. Subspace methods suffer from cubic time complexity: specifically, the original ESPRIT algorithm relies on obtaining the signal's singular vectors by SVD/EVD. We show that the original ESPRIT algorithm does not need the signal's singular vectors to exploit rotational invariance, but a set of vectors that span the signal model's Vandermonde matrix are sufficient. To exploit this, we introduce two FFT-based ESPRIT algorithms which have lower time complexities than the original ESPRIT algorithm. These fast algorithms rely on the usage of the DFT kernel, i.e.~a Fourier transform on the row space of the data (Hankel) matrix. The preferred method, named FFT-ESPRIT, achieves quasi-linear time complexity due to a fast Hankel matrix-Kernel matrix product algorithm and extensive usage of the FFT. The kernel-based subspace estimates are approximate in nature, and yet perturbation bounds reveal a noise regime in which FFT-ESPRIT exceeds ESPRIT's performance. We demonstrate the behavior of the algorithm across different SNR and super-resolution regimes and show that the estimated signal subspace is statistically efficient. Numerical simulations show that FFT-ESPRIT is more robust than the ESPRIT algorithm at the very low SNRs, and has a nearly identical performance as ESPRIT at higher SNRs.