  Subspace-based signal processing techniques, such as the Estimation of Signal
Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular
methods for spectral estimation. These algorithms can achieve the so-called
super-resolution scaling under low noise conditions, surpassing the well-known
Nyquist limit. However, the performance of these algorithms under high-noise
conditions is not as well understood. Existing state-of-the-art analysis
indicates that ESPRIT and related algorithms can be resilient even for signals
where each observation is corrupted by statistically independent, mean-zero
noise of size $\mathcal{O}(1)$, but these analyses only show that the error
$\epsilon$ decays at a slow rate $\epsilon=\mathcal{\tilde{O}}(n^{-1/2})$ with
respect to the cutoff frequency $n$. In this work, we prove that under certain
assumptions of bias and high noise, the ESPRIT algorithm can attain a
significantly improved error scaling $\epsilon =
\mathcal{\tilde{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling
beyond the Nyquist limit. We further establish a theoretical lower bound and
show that this scaling is optimal. Our analysis introduces novel matrix
perturbation results, which could be of independent interest.
