around those produced in the first iteration. As a result, the third spike is identified while the first two are still not. Following from the same reweighting process, all the frequencies are correctly determined in the next iteration and the algorithm converges after that. It is worth noting that $T\sbra{\m{u}^*}$ becomes rank-5 and the remaining eigenvalues become zero (within numerical precision) after three iterations, where $\epsilon=0.25$. Finally, we report that the relative mean squared error (MSE) of signal recovery improves during the iterations from $1.28\times10^{-4}$ to $2.01\times 10^{-7}$, $3.16\times10^{-19}$ and $3.01\times10^{-21}$. Each iteration takes about 1.7s.

![image](Fig_illusexample.pdf){width="6in"}

Sparsity-Separation Phase Transition
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In this subsection, we study the success rate of RAM in signal and frequency recovery compared to ANM. In particular, we fix $N=64$ and $M=30$ with the sampling index set $\m{\Omega}$ being generated uniformly at random. We vary the duo $\sbra{K,\Delta_f}$ and for each combination we randomly generate $K$ frequencies such that they are mutually separated by at least $\Delta_f$. We randomly generate the amplitudes $\lbra{s_{kt}}$ independently and identically from a standard complex normal distribution. After obtaining the noiseless samples, we carry out signal reconstruction and frequency recovery using ANM and RAM, both implemented by SDPT3. The recovery is 