  We consider the problem of positioning a cloud of points in the Euclidean
space $\mathbb{R}^d$, using noisy measurements of a subset of pairwise
distances. This task has applications in various areas, such as sensor network
localization and reconstruction of protein conformations from NMR measurements.
Also, it is closely related to dimensionality reduction problems and manifold
learning, where the goal is to learn the underlying global geometry of a data
set using local (or partial) metric information. Here we propose a
reconstruction algorithm based on semidefinite programming. For a random
geometric graph model and uniformly bounded noise, we provide a precise
characterization of the algorithm's performance: In the noiseless case, we find
a radius $r_0$ beyond which the algorithm reconstructs the exact positions (up
to rigid transformations). In the presence of noise, we obtain upper and lower
bounds on the reconstruction error that match up to a factor that depends only
on the dimension $d$, and the average degree of the nodes in the graph.
