Lifted Restricted Isometry Property and Fast Algorithm for Spectral Compressed Sensing With Near-Optimal Sample Complexity

This paper studies the spectral compressed sensing problem, which aims to recover an $r$-spectrally-sparse signal of length $N$ from linear compressive measurements. We reformulate the problem as the recovery of a low-rank Hankel matrix. Motivated by this formulation, we propose a Lifted Restricted Isometry Property (LRIP) for the measurement matrix and show that Gaussian measurement matrices satisfy LRIP under certain conditions. Unlike classical RIP, LRIP imposes a rank constraint in the lifted matrix domain while taking measurements in the signal domain through the adjoint Hankel operator. To solve the resulting nonconvex problem efficiently, we introduce a Fast Iterative Hard Thresholding (FIHT) algorithm. Based on LRIP, we show that FIHT converges linearly to the true signal with high probability using $\mathcal{O}(r\ln r{\ln }^{2}N)$ scaled Gaussian measurements, which is near-optimal up to log factors and improves the state-of-the-art order of $r^2$ for nonconvex Hankel methods. Extensive numerical results demonstrate the superiority of FIHT as compared to state-of-the-art convex methods in terms of phase transitions, computational time, and robustness to noise.