Material for “ Spectral Compressed Sensing via Projected Gradient Descent ”
نویسندگان
چکیده
We extend PGD and its recovery guarantee [1] from one-dimensional spectrally sparse signal recovery to the multi-dimensional case. Assume the underlying multi-dimensional spectrally sparse signal is of model order r and total dimension N . We show that O(r log(N)) measurements are sufficient for PGD to achieve successful recovery with high probability provided the underlying signal satisfies some incoherence property. 1 Algorithm and Main Result Without loss of generality, we discuss the two-dimensional setting but emphasize that the situation in general d-dimensions is similar. Let wk = e (2πıf1k−τ1k) and zk = e (2πıf2k−τ2k) for r frequency pairs (f1k, f2k) ∈ [0, 1)2 and r damping factor pairs (τ1k, τ2k) ∈ R+. A two-dimensional spectrally sparse array X ∈ CN1×N2 can be expressed as X = r ∑ k=1 dkw a kz b k, (a, b) ∈ [N1]× [N2]. The two-fold Hankel matrix of X is given by HX = HX(:,0) HX(:,1) HX(:,2) · · · · · · HX(:,N2−n2) HX(:,1) HX(:,2) · · · · · · · · · HX(:,N2−n2+1) HX(:,2) · · · · · · · · · · · · HX(:,N2−n2+2) .. .. .. .. .. .. HX(:,n2−1) HX(:,n2) · · · · · · · · · HX(:,N2−1) , Email addresses: [email protected] (J.-F. Cai), [email protected] (T. Wang), and [email protected] (K. Wei, corresponding author).
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