Sensor network localization has a benign landscape after low-dimensional relaxation

We consider the sensor network localization problem, which is closely related to multidimensional scaling and Euclidean distance matrix completion. Given a ground truth configuration of $n$ points in $\mathbb{R}^\ell$, we observe a subset of the pairwise distances and aim to recover the underlying configuration (up to rigid transformations). We show with a simple counterexample that the associated optimization problem is nonconvex and may admit spurious local minimizers, even when all distances are known. Yet, inspired by numerical experiments, we argue that all second-order critical points become global minimizers when the problem is relaxed by optimizing over configurations in dimension $k>\ell$. Specifically, we show this for two settings, both when all pairwise distances are known: (1) for arbitrary ground truth points, and $k= O(\sqrt{\ell n})$, and: (2) for isotropic random ground truth points, and $k = O(\ell + \log n)$. To prove these results, we identify and exploit key properties of the linear map which sends inner products to squared distances.