Subspace Knowledge for Spectral Compressed Sensing

We consider the problem of recovering a spectrally sparse signal from its partially observed entries. Traditional approaches leverage the low-rank property of such signals in the Hankel lifting domain. In this domain, we further incorporate subspace knowledge, initially estimated from test data and updated during signal recovery, to enhance performance. To quantify the impact of subspace knowledge on this enhancement, we propose a metric that integrates QR decompositions with the subspace direction number to assess its accuracy, as reflected in the recovery guarantee. The established guarantee demonstrates the local optimality of the proposed nonconvex method for signal recovery. Based on the local optimality, the proposed method achieves exact recovery with high probability, provided that the initial subspace estimate is sufficiently accurate to meet certain conditions. Compared to prior methods without subspace knowledge and those that exploit such knowledge, our method reduces the sample complexity by three and two logarithmic factors, respectively. Furthermore, we employ low-rank factorization techniques to incorporate prior information of the Hankel matrix rank, thereby enhancing the recovery precision and computational efficiency of the proposed method. The proposed method is efficiently optimized in an alternating minimization framework. Experimental results demonstrate the superior recovery precision and reduced sample complexity of the proposed method compared to existing methods, as well as its enhanced performance through low-rank factorization.