  Motivated by a number of applications in signal processing, we study the
following question. Given samples of a multidimensional signal of the form
\begin{align*} f(\bs\ell)=\sum_{k=1}^K a_k\exp(-i\langle \bs\ell, \w_k\rangle),
\\ \w_1,\cdots,\w_k\in\mathbb{R}^q, \ \bs\ell\in \ZZ^q, \ |\bs\ell| <n,
\end{align*} determine the values of the number $K$ of components, and the
parameters $a_k$ and $\w_k$'s. We develop an algorithm to recuperate these
quantities accurately using only a subsample of size $\O(qn)$ of this data. For
this purpose, we use a novel localized kernel method to identify the
parameters, including the number $K$ of signals. Our method is easy to
implement, and is shown to be stable under a very low SNR range. We demonstrate
the effectiveness of our resulting algorithm using 2 and 3 dimensional examples
from the literature, and show substantial improvements over state-of-the-art
techniques including Prony based, MUSIC and ESPRIT approaches.
