Supplementary Material for Fast and Provable Algorithms for Spectrally Sparse Signal Reconstruction via Low-Rank Hankel Matrix Completion

We establish theoretical recovery guarantees of FIHT for multi-dimensional spectrally sparse signal reconstruction problems, which are straightforward extensions of what we have proved for one-dimensional signals in [1]. Assume the underlying multi-dimensional spectrally sparse signal is of model order r and total dimensionN . We show that O(r log(N)) number of measurements are sufficient for FIHT with resampling initialization to achieve reliable reconstruction provided the signal satisfies the incoherence property. 1 Recovery Guarantees Without loss of generality, we discuss the three-dimensional setting. Recall that a three-dimensional array X ∈ CN1×N2×N3 is spectrally sparse if X (l1, l2, l3) = r ∑ k=1 dky l1 k z l2 k w l3 k , ∀ (l1, l2, l3) ∈ [N1]× [N2]× [N3] with yk = exp(2πıf1k − τ1k), zk = exp(2πıf2k − τ2k), and wk = exp(2πıf3k − τ3k) for frequency triples fk = (f1k, f2k, f3k) ∈ [0, 1)3 and dampling factor triples τk = (τ1k, τ2k, τ3k) ∈ R+. Concatenating the columns of X, we get a signal x of length N1N2N3. Define N = N1N2N3. We form a three-fold Hankel matrix Hx, which has Vandermonde decomposition in the form Hx = ELDE T R, where the k-th columns (1 ≤ k ≤ r) of EL and ER are given by E (:,k) L = { y1 k z l2 k w l3 k , (l1, l2, l3) ∈ [p1]× [p2]× [p3] } , E (:,k) R = { y1 k z l2 k w l3 k , (l1, l2, l3) ∈ [q1]× [q2]× [q3] } , ∗Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, China. Email: [email protected] †Department of Mathematics, University of Iowa, Iowa City, Iowa, USA. Email: [email protected] ‡Department of Mathematics, University of California, Davis, California, USA. Email: [email protected]