Before revision:  In particular, our fast MUST algorithms speed up the expensive signal subspace identification step using the randomized low-rank approximations. We also establish theoretical bounds to guarantee the quality of approximation. Specifically, we show that, in the case of high signal-to-noise ratio, the pseudo-spectrum produced by our algorithms are very close to the exact MUSIC.
After revision:  In particular, our fast MUST algorithms speed up the subspace computation by orders of magnitude. At the same time, we establish the theoretical bounds of our proposed approaches, which ensure the accuracy of approximated pseudo-spectrum. As shown, in the case of high signal-to-noise ratio, the pseudo-spectrum produced by our algorithms are very close to the exact MUSIC.