  We address the problem of reconstructing a set of points on a line or a loop
from their unassigned noisy pairwise distances. When the points lie on a line,
the problem is known as the turnpike; when they are on a loop, it is known as
the beltway. We approximate the problem by discretizing the domain and
representing the $N$ points via an $N$-hot encoding, which is a density
supported on the discretized domain. We show how the distance distribution is
then simply a collection of quadratic functionals of this density and propose
to recover the point locations so that the estimated distance distribution
matches the measured distance distribution. This can be cast as a constrained
nonconvex optimization problem which we solve using projected gradient descent
with a suitable spectral initializer. We derive conditions under which the
proposed distance distribution matching approach locally converges to a global
optimizer at a linear rate. Compared to the conventional backtracking approach,
our method jointly reconstructs all the point locations and is robust to noise
in the measurements. We substantiate these claims with state-of-the-art
performance across a number of numerical experiments. Our method is the first
practical approach to solve the large-scale noisy beltway problem where the
points lie on a loop.
