Exact Dimensionality Reduction for Partial Line Spectra Estimation Problems

Line spectral estimation theory aims to estimate the off-the-grid spectral components of a time signal with optimal precision. Recent results have shown that it is possible to recover signals having sparse line spectra from few temporal observations via the use of convex programming. However, the computational cost of such approaches remains the major flaw to their application to practical systems. This work investigates the recovery of spectrally sparse signal from low-dimensional partial measurements. It is shown in the first part of this paper that, under a light assumption on the sub-sampling matrix, the partial line spectral estimation problems can be relaxed into a low-dimensional semidefinite program. The proof technique relies on a novel extension of the Gram parametrization to subspaces of trigonometric polynomials. The second part of this work focuses on the analysis of two particular sub-sampling patterns: multirate sampling and random selection sampling. It is shown that those sampling patterns guarantee perfect recovery of the line spectra, and that the reconstruction can be achieved in a poly-logarithmic time with respect to the full observation case. Moreover, the sub-Nyquist recovery capabilities of such sampling patterns are highlighted. The atomic soft thresholding method is adapted in the presented framework to estimate sparse spectra in noisy environments, and a scalable algorithm for its resolution is proposed.