Super-resolution of near-colliding point sources

Gil Goldman, Dmitry Batenkov, Yosef Yomdin

Abstract:

We consider the problem of stable recovery of sparse signals of the
form
$$F(x)=\sum_{j=1}^d a_j\delta(x-x_j),\quad x_j\in\mathbb{R},\;a_j\in\mathbb{C}, $$
from their spectral measurements, known in a bandwidth $\Omega$ with
absolute error not exceeding $\epsilon>0$. We consider the case
when at most $p\le d$ nodes $\{x_j\}$ of $F$ form a cluster whose
extent is smaller than the Rayleigh limit ${1\over\Omega}$, while
the rest of the nodes are well separated. Provided that
$\epsilon \lessapprox \operatorname{SRF}^{-2p+1}$, where $\operatorname{SRF}=(\Omega\Delta)^{-1}$
and $\Delta$ is the minimal separation between the nodes, we show
that the minimax error rate for reconstruction of the cluster nodes
is of order ${1\over\Omega}\operatorname{SRF}^{2p-1}\epsilon$, while for recovering
the corresponding amplitudes $\{a_j\}$ the rate is of the order
$\operatorname{SRF}^{2p-1}\epsilon$. Moreover, the corresponding minimax rates for
the recovery of the non-clustered nodes and amplitudes are
${\epsilon\over\Omega}$ and $\epsilon$, respectively. These results
suggest that stable super-resolution is possible in much more
general situations than previously thought. Our numerical
experiments show that the well-known Matrix Pencil method achieves
the above accuracy bounds.