Motivated by a number of applications in signal processing, we study the following question. Given samples of a multidimensional signal of the form $$ f(\boldsymbol\ell)=\sum_{k=1}^K a_k\exp(-i\langle \boldsymbol\ell, \mathbf{w}_k\rangle), \quad \mathbf{w}_1,\cdots,\mathbf{w}_k\in\mathbb{R}^q, \ \boldsymbol\ell\in \mathbb{Z}^q, \ |\boldsymbol\ell| <n, $$ determine the values of the number $K$ of components, and the parameters $a_k$ and $\mathbf{w}_k$'s. We note that the the number of samples of $f$ in the above equation is $(2n-1)^q$. We develop an algorithm to recuperate these quantities accurately using only a subsample of size $\mathcal{O}(qn)$ 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.